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Presented to the 



by the 

ONTARIO LEGISLATIVE 
LIBRARY 

1980 



NAVIGATION 



THE MACMILLAN COMPANY 

NEW YORK BOSTON CHICAGO DALLAS 
ATLANTA SAN FRANCISCO 

MACMILLAN & CO., LIMITED 

LONDON BOMBAY CALCUTTA 
MELBOURNE 

THE MACMILLAN CO. OF CANADA, LTD. 

TORONTO 



NAVIGATION 




BY 

HAROLD JACOBY 

RUTHERFURD PROFESSOR OF ASTRONOMY 
IN COLUMBIA UNIVERSITY 



SECOND EDITION 

WITH A CHAPTER ON COMPASS ADJUSTING AND A 
COLLECTION OF MISCELLANEOUS EXAMPLES 



ELECTRONIC VERSION 
AVAILABLE 



NO. 



THE MACMILLAN COMPANY 




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COPYKIGHT, 1917 AND 1918, 

BY THE MACMILLAN COMPANY. 



Set up and electrotyped. Published October, 1917. 



Second edition, with new matter, February, 1918. 



Norfoooti 

J. S. Gushing Co. Berwick & Smith Co. 
Norwood, Mass., U.S.A. 



Co 
MACLEAR JACOBY 

QUARTERMASTER,* THIRD CLASS, U. 8. N. 

BNLI8TED FOR THE PERIOD OF THE WAR 

THIS VOLUME IS OFFERED AS 

A MARK OF RESPECT 

BY HIS FATHER 



* COMMISSIONED ENSIGN, U. S. N. R. F., SEPTEMBER, 1917 



THE present volume was undertaken with certain very 
definite aims. In the first place, it is intended to be com- 
plete in itself, so that it should be possible to navigate a ship 
in any ocean not very near the north or south pole without 
other books or tabular works, excepting only the nautical 
almanac for the year in which the voyage is made. To attain 
this end without unduly extending the size of the volume, 
certain essential nautical tables have been abridged; but 
all are given in sufficiently extended form to permit of actual 
navigation with their aid; and they are especially suitable 
for beginners, who can here attain the necessary knowledge 
with less effort than would be necessary with more bulky 
volumes. In cases where very extended tables are conven- 
ient, they are mentioned in the text. 

In the second place, the author has not assumed that the 
reader possesses formal mathematical and astronomical 
knowledge, or desires to possess such knowledge. When- 
ever methods of navigation require for their demonstration 
an understanding of spherical trigonometry, or some other 
branch of formal mathematical science, such demonstrations 
have been replaced with incomplete or "outline " demonstra- 
tions designed for the non-mathematical reader. Practical 
methods are fully explained ; and an attempt has always 
been made so to word the explanations that the reader, 
even the beginner, will understand his problem, and will 
know what he is doing, and why he does it. 

The requirements of those who may study without a 
teacher have received constant and special attention. To 
meet these requirements the whole subject is presented in 

vii 



viii PREFACE 

a somewhat informal manner; such topics as the use of 
logarithms, or the principles on which all mathematical 
tables are constructed these less attractive parts of the 
subject are not presented in a special chapter, but are de- 
scribed in a sort of digression, when needed in the discussion 
of an actual navigational problem. 

Finally, to further simplify and condense his material, 
the author has made no attempt to include every method 
that can possibly be used to navigate a ship, or that ever has 
been used to navigate a ship ; his purpose has been rather 
to limit the volume to the methods at present thought best 
by the most reliable modern authorities. 

Other books on navigation have been used freely, espe- 
cially in the preparation of the tables. Among these, that 
admirable encyclopedia of navigation, known as "Bowditch," 
published by the Hydrographic Office, United States Navy, 
and Kelvin's "Tables for Sumner's Method at Sea" have 
been found of the greatest help. 

Miss Dorothy W. Block, Instructor of Astronomy in 
Hunter College, New York, has helped with great energy 
in the preparation of the tables and the correction of the 
text. It is hoped that suph errors as may now remain in 
the book are few in number. 

H. J. 

COLUMBIA UNIVERSITY, 
August, 1917. 



PREFATORY NOTE TO THE SECOND EDITION 

To meet the wishes of certain young navigators, this edition 
has an added chapter on the adjustment of correctors in a 
compensated compass binnacle, and also a collection of new 
problems and examples. 

H. J. 

February, 1918. 



TABLE OF CONTENTS 

HAPTEK PAGH 

I. THE FUNDAMENTAL PROBLEM OF NAVIGATION . . 1 

The problem stated. Reasons for the existence of the 
problem. Definition of "ship's position." Longitude 
meridians and latitude parallels. Greenwich the initial 
meridian. Position determined by observation; on the 
coast and at sea. Dead reckoning. Sextant observa- 
tions. Chronometer. 

II. DEAD RECKONING WITHOUT LOGARITHMS . . > .,.: .7 

The two problems. Designation of .ship's course. 
Latitude difference and departure. The traverse table. 
Use and construction of tables in general. Arguments 
and tabular numbers. Relation between departure and 
longitude difference. Middle latitude. 

III. DEAD RECKONING WITH LOGARITHMS ... 23 
Explanation of number logarithms and their use. 

Multiplication and division. Trigonometric logarithms. 
Solution of the two problems. Middle latitude sailing. 
Mercator sailing. Meridional parts. Great circle sail- 
ing. The rhumb line. Composite sailing. Parallel 
sailing. Traverse sailing. 

IV. THE COMPASS 40 

The card, how divided. Degrees and points. Boxing 

the compass. Lubber line. True course and compass 
course. Error, variation and deviation. Swinging ship. 
Azimuth circle and pelorus. The compass formulas. 
The two deviation tables. Comparative table of points 
and degrees. 

V. COASTWISE NAVIGATION ...... 53 

The "fix." Bow bearings. Doubling the bearing on 
the bow. Bow and beam bearings. Distance a-beam. 
Cross bearings. The danger angle. Danger bearing. 
Soundings. 

ix 



X TABLE OF CONTENTS 

CHAPTER PAGE 

VI. THE SEXTANT . 61 

Description of the instrument and its use. The vernier. 
Index error. Three adjustments. The artificial horizon. 
Correcting the altitude. Dip. Refraction. Parallax. 

VII. THE NAUTICAL ALMANAC ...... 75 

Specimen pages of it. Greenwich mean time. Decli- 
nation. Equation of time. Astronomic and civil day. 
Apparent solar time. Chronometers and the rate card. 
Right ascension. Solar and sidereal time. 

VIII. OLDER NAVIGATION METHODS 86 

The noon-sight for latitude. Tropic observations and 
the midnight sun in high latitudes. Preparing for the 
observation. Setting the cabin clock. Star observa- 
tion. Ex-meridian observation. The time-sight for 
longitude. Set of current. Star time-sight. Condensed 
forms of calculation. 

IX. NEWER NAVIGATION METHODS ..... 108 

Errors produced by dead reckoning. Captain Sumner, 
and the Sumner line. Bearing of the line. The Sumner 
point. Azimuth tables. Condensed form of calcula- 
tion. Star observations. Comparison of Sumner navi- 
gation with time-sight navigation. The Kelvin table. 
Condensed forms of sun and star observations. Inter- 
section of two Sumner lines obtained with a special table. 
Motion of ship between observations. 

X. A NAVIGATOR'S DAY AT SEA ..... 141 

Voyage planned from New York to Colon. Departure 
at Sandy Hook lightship. The course to Watlings Island. 
The variation and deviation applied. Azimuth of the sun 
observed at sunrise. Bow and beam bearings of Barnegat 
Light. The patent log and the log book. New course 
from Barnegat. Morning sight worked as a Sumner line. 
Another Azimuth observation. Weather thickens at 
11 : 30. Ex-meridian sight at 11 : 42, worked as a Sumner 
line. Afternoon sight worked as a Sumner line. Posi- 
tion of ship fixed from intersection of the two lines. East- 
erly current estimated. Compass error again tested. 
The course set for the night. 
TABLES . . . . . . . . . .153 

APPENDIX 1. Compass Adjusting . . . . . 323 

APPENDIX 2. Miscellaneous Examples ..... 335 



LIST OF ABBREVIATIONS 

USED IN THE PRESENT VOLUME 

Alt. for altitude ; 

App. for apparent ; 

Arg. diff . for argument difference ; 

Cf . for compare ; 

Chron. for chronometer ; 

Comp'd for computed ; 

Cos for cosine ; 

Cot for cotangent ; 

Csc for cosecant ; 

C. W. for chronometer minus watch ; 
Dec. for declination ; 

Dep. for departure ; 

Dist. for distance ; 

D. R. for dead reckoning; 
Eq. for equation of time ; 

G. A. T. for Greenwich apparent time; 

G. M. T. for Greenwich mean time ; 

Hav. for haver sine ; 

H. D. for hourly difference ; 

Int. diff. for interpolation difference ; 

Lat. for latitude ; 

Lat. diff. for latitude difference ; 

Log for logarithm ; 

Long. for longitude ; 

Long. diff. for longitude difference ; 

Mer. lat. diff. for meridional latitude difference ; 

Obs'd for observed ; 

p for polar distance ; 

R. A. for right ascension ; 

s for hah 3 sum ; 

Sec for secant ; 

Sin for sine ; 

T for ship's apparent solar time (or star's hour-angle) ; 

Tab. diff. for tabular difference ; 

Tan for tangent. 

xi 



NAVIGATION 

CHAPTER I 
THE FUNDAMENTAL PROBLEM OF NAVIGATION 

To find one's way in a ship across the trackless ocean is 
our problem. Most people would like to know how it is 
solved ; nor is the solution very difficult to understand when 
set forth in simple language and without too great wealth of 
technical detail. We hope the reader will find this to be 
the case after a study of the following pages. 

Our fundamental problem can be more fully stated quite 
easily. It consists in the determination of a ship's location 
on the earth's surface at any given moment. If this loca- 
tion can be determined, it becomes a comparatively easy 
matter to ascertain the direction (north, south, northeast, 
southeast, etc.) in which the ship must be steered in order 
to reach her port of destination. For the location of the 
port of destination on the earth's surface is of course also 
known : and if we know where the ship and her destined port 
both are, we can easily find the right course for the helmsman. 

With the fundamental problem stated in this way, it 
would almost seem as if there were really no such problem 
in existence. For when the ship begins her voyage, she is 
necessarily in a known port. Knowing also the port to 
which she is to go, we should be able to determine her proper 
course from the one known port to the other. This course 
being then steered, no further navigational proceedings would 
be required. But this reasoning is incorrect, because a ship 
B 1 



2 NAVIGATION 

does not actually advance across the ocean in exactly the 
direction in which she is steered. Ocean currents deflect 
her ; and the action of a strong wind blowing against one of 
her sides will have a similar effect. Currents and winds 
cannot be predicted with accuracy : and so it becomes 
necessary to re-determine the ship's position frequently at 
sea. This should be done at least once daily if possible; 
and when it has been done, the mariner can take a new 
"departure," as he calls it, and lay a new course for his 
intended port. Thus the effect of ocean currents, etc., can 
be eliminated, and the voyage made as safely as if they did 
not exist. 

Now this determination of the ship's position at sea, 
and when out of sight of land, is strictly an astronomical 
problem. It can be solved by means of astronomical ob- 
servations, and in no other way. But before giving an out- 
line of how this is done, let us first see what is meant by 
the words "ship's position at sea." How can we describe 
a ship's position so that one mariner could tell another 
where she is located, and thus enable the second mariner to 
find her? 

To thus indicate the point on the earth's surface occupied 
by the ship has a certain similarity with giving the address of 
a house in a city. Such a city address always consists of 
two separate statements; as, for instance, the name of a 
street and the number of the house. An address cannot 
be given completely unless two different facts are stated. 
They need not necessarily be a street name and a street 
number : we can equally well designate such an address by 
stating that the house is at the corner of a certain street and 
a certain avenue. But here also the address is made up of 
two separate facts. 

This form of stating an address as the intersection of a 

certain street and avenue is the form having the closest 

^resemblance to the method of the navigator. If the city 

avenues are supposed to run north and south, and the streets 



THE FUNDAMENTAL PROBLEM OF NAVIGATION 3 

east and west, as they do in New York (approximately), the 
analogy with navigation will be almost perfect. 

For the navigator imagines the earth covered with a net- 
work consisting of "avenues, " running north and south, and 
"streets," running east and west. He calls the "avenues" 
meridians of longitude, and the " streets " parallels of latitude. 
Then he designates the position of a ship on the ocean by 
stating that it is at the intersection of a certain meridian 
of longitude and parallel of latitude. There are 360 such 
meridians of longitude : each begins at the terrestrial equator, 
and runs north and south from there to the north and south 
poles of the earth. Of the latitude parallels there are ISO. 1 
They all run east and west, parallel to the terrestrial equator ; 
90 are between the equator and the north pole, and the other 
90 between the equator and the south pole. 

One of the longitude meridians (that passing through 
Greenwich, England) is chosen arbitrarily as the starting 
point for counting longitude meridians. To this initial 
meridian is assigned the number 0, and the other meridians 
are numbered successively 1, 2, 3, etc. So numbered, 
the meridians are called "degrees" of longitude; the third 
one, for instance, being written 3. The meridians may be 
counted either eastward or westward from Greenwich, a 
ship on the 20th meridian west of Greenwich, for instance, 
being in longitude 20 west. 

The latitude parallels are similarly counted north and 
south from the equator ; and if the above ship were on the 
40th latitude parallel north of the equator, her complete 
"address," or position at sea, would be long. 20 W. ; lat. 
40 N. 

Of course a ship would only rarely be located exactly at 
the intersection of a meridian and parallel. Therefore, the 
space between any two successive meridians and between 
any two successive parallels is subdivided into 60 parts, 
called minutes of arc. Thus the above ship, if halfway 
1 Including the equator twice, but excluding the two poles. 



4 NAVIGATION 

between a pair of meridians and also halfway between a 
pair of parallels, might be in longitude 20 30' west, and 
in latitude 40 30' north. This would be written long. 
20 30' W. ; lat. 40 30' N. 

Each minute of longitude and latitude is further sub- 
divided, when extreme accuracy is required, into 60 seconds ; 
so that if the ship were a little to the north and a little to 
the west of the above position, she might, for instance, be 
in long. 20 30' 26" W. ; lat. 40 30' 10" N. 

These meridians and parallels, or longitude and latitude 
lines, appear on many maps and charts as straight lines, 
or at least as lines only slightly curved. But being all lines 
imagined drawn on the earth, which is almost an exact 
sphere or round ball, they must really all be circles. Thus, 
the terrestrial equator is really a big circle, girdling the 
earth, and divided into 360 equal parts, or degrees. At 
each of the division points a meridian starts northward 
toward the pole. This meridian is also a big circle 
perpendicular to the equator. The distance along the 
meridian from the equator to the pole is divided into 90 
equal parts or degrees, and the whole distance from equator 
to pole is one quarter of a complete circumference of the 
earth. The 90 degrees, from equator to pole, thus repre- 
senting one quarter of a circumference of the earth, a com- 
plete circumference contains 4 X 90, or 360 degrees, the 
same as the equator. So the degrees measured along the 
meridians are equal to the degrees measured along the 
equator. The former are degrees of latitude, the latter 
degrees of longitude; and degrees of latitude are equal to 
degrees of longitude, when the latter are measured along 
the equator. The length of each degree is then 60 nautical 
miles. 

Having thus indicated what is meant by a ship's position 
in latitude and longitude, we shall next describe in outline 
how such a position may be determined by observation. 
Tf the ship is within sight of a coast-line, there will probably 



THE FUNDAMENTAL PROBLEM OF NAVIGATION 5 

be some lighthouse, or other "aid to navigation," in view, 
from which the navigator can ascertain where he is. Methods 
for doing this are described later (p. 53). But when -the 
ship is really at sea, with no land in sight, real deep-sea 
methods must be employed. 

These methods, when the weather is clear, always include 
an observation of the sun or some other heavenly body. 
When the weather does not permit such observations, the 
mariner can still find his position approximately by means 
of "dead reckoning" (abbreviated, D. R.). This process 
will be described in detail in the next chapter; but we can 
already state that it consists in a calculation based on his 
astronomic observation of latest date. Knowing where the 
ship was the last time he observed the sun, and -also know- 
ing both the direction in which he has steered and the 
(approximate) speed of the ship, the navigator can calculate 
(also approximately) the location of the point he has reached. 

Even when astronomical observations are made, the 
D. R. calculation is always carried out, because the navi- 
gator is always anxious to know how nearly correct his 
D. R. result would have been, if the day had been cloudy. 
Furthermore, this result also acts as a check on the astronomi- 
cal work, and tends to increase the navigator's confidence 
in the correctness of his final result as to the ship's location. 

The manner in which the ship's position is found from 
astronomic observations will of course be explained in detail 
later. It is all done with an instrument called a sextant. 
This is merely a contrivance with which the navigator can 
measure how high the sun (or other heavenly body) is in the 
sky at any moment. The sun is highest in the sky daily 
at noon, but it is not equally high on different days in the 
year. Nor is it equally high on the same date in different 
latitudes. Thus, by measuring with the sextant how high 
it is on any particular date at noon, as seen from the ship, 
the navigator learns the terrestrial latitude in which the 
ship is located. 



6 NAVIGATION 

Similar sextant observations made at other suitable times 
during the day, when combined with exact readings taken 
from an accurate chronometer such as every ocean-going 
ship carries, will similarly make the ship's longitude known, 
All this will of course be explained in full detail in later 
chapters. 



CHAPTER II 
DEAD RECKONING WITHOUT LOGARITHMS 

As we have seen (p. 5), this is a process by means of 
which the mariner can calculate a ship's position in latitude 



r e 


\ 

0' 5 


/Vest Lc 

9 5 


jngitud* 
8 5 


\ 

r 5 


6 5 


sr 

46 














45 














44' 




NJ 


s/r P 
W/-/ 

/o / 








43 




1 


/ 








42 














41' 














40' 

















Fio. 1. Dead Reckoning. (Diagram not drawn to scale.) 
7 



8 NAVIGATION 

and longitude, without special astronomic observations of 
any kind. In the accompanying Fig. 1, which represents a 
portion of a chart of the North Atlantic, a ship's position 
at noon is shown at the point Y. This point we will call 
the ship's "initial position," in discussing our present prob- 
lem. We will suppose that it was correctly obtained by as- 
tronomic observations, and that these showed the ship at Y 
to be in lat. 42 11' N. and long. 59 28' W. from Green- 
wich. Sometime in the afternoon, having traveled a dis- 
tance estimated from the known speed of the ship as 63 miles, 
and having "made good" this distance in the direction YP, 
the ship arrives at P. This point P we will call the ship's 
"final position" ; and our problem now is to find its latitude 
and longitude. 

This problem may be called the first fundamental dead- 
reckoning problem. The second and remaining fundamental 
problem is the converse of the first, and may be stated as 
follows : having given the latitude and longitude of the initial 
point Y, as occupied by the ship, and also the latitude and 
longitude of the final point P, it is required to find the dis- 
tance from Y to P in miles, and also the direction of the line 
YP. 1 

To understand these two problems properly it is next 
necessary to explain how we may define the words "direc- 
tion YP." This is done by referring the line YP to the 
direction of the arrow shown in the figure. This arrow 
is parallel to the longitude meridians on the chart, and 
therefore points due north. The angle between the arrow 
YN and the line YP is marked in the figure, and is called 
the "ship's course." This angle is really the difference in 
direction of the two lines YN and YP. The point Y is called 
the "vertex" of the angle, and all angles are designated 

1 We think it advisable to place these two important converse 
problems together, and to call them both pro.blems^of. dead reckon- 
ing, though many writers on navigation confine the phrase " dead 
reckoning" to the first fundamental problem alone. 



DEAD RECKONING WITHOUT LOGARITHMS 9 




FIG. 2. Dead 
Reckoning. 



by three letters, the letter belonging to the vertex being 
placed between the other two; in this case the angle is 
called either NYP or PYN. 

Now let us draw a line PQ (fig. 2), from P to NY, and 
perpendicular to NY. Then the motion of the ship from 
Y to P will have carried her north of the N,, 
point Y by a distance equal to YQ, and east 
of the point Y by a distance equal to QP. Q 
This is not strictly true, unless the earth's 
surface, throughout the small area involved 
in the present problem, can be regarded 
as a flat surface. Such a flat surface is 
called in geometry a "plane" surface; and 
these calculations therefore belong to that 
part of navigation which is called "plane sailing." Plane- 
sailing calculations are easy calculations, and they are 
generally sufficiently accurate for the purposes of the 
navigator. 

The ship's course, being thus an angle, must be designated 

by means of a unit of measure 
suitable for measuring angles. 
For this purpose the degrees and 
minutes already used for longi- 
tude and latitude (p. 3) are 
usually employed. Fig. 3 shows 
that a latitude, for instance, is 
really an angle, and must there- 
fore also be measured in de- 
grees. P is the earth's pole, PQ 
a meridian, and the latitude of 
the observer at is the angle 
OCQ, here about 40. 
So it is clear that the ship's course NYP (figs. 1 and 2) 
will be measured in degrees. Minutes are not really needed 
in measuring courses, as they are in measuring latitudes; 
the nearest whole degree is always accurate enough, because 




FIG. 3. Latitude Angle. 




10 NAVIGATION 

it is never possible to steer a ship on her proper course with 
absolute exactness. In fact, many mariners use a still less 
precise method of measuring courses by means of "the points 
of the compass." (See p. 40.) 

Resuming our two fundamental problems (p. 8), let us 
now begin with the first one, and proceed to find the lati- 
tude and longitude of the point P (figs. 1 and 2). To solve 
this problem, we must not only know the distance YP 
(63 miles), as traveled by the ship, but also the number of 
degrees in the course angle NYP. Let us suppose this course 
angle happens also to be 40. The problem 
then appears as shown in Fig. 4. We now 
know the distance YP and the angle QYP. 
Evidently the next step is to find the distances 
QFand QP. QY, in our present problem, is 
called a "latitude difference" and QP is called 
a "departure." 
FIG. 4. -Dead To find the "latitude difference" and 
"departure" from the course angle and dis- 
tance we may either use that branch of mathematics called 
plane trigonometry, or we may find them from a special 
navigation table, called a "traverse table." Our Table 1 
(beginning p. 154) is such a table. 

Before l beginning its use it will be well for the reader to 
note in general that all mathematical tables consist of two 
sets of numbers. The first set of numbers are called " argu- 
ments" of the table, and the second set are called "tabular 
numbers." The main object of the table is to furnish us 
with the proper tabular number when we know the proper 
argument. 

The ordinary multiplication table is a good example of a 
mathematical table. It is usually .written as follows and 

1 The beginner may find it advisable, on a first reading of the 
book, to omit this explanation of mathematical tables, returning 
later when he finds a reference to it in the text. The dead reckoning 
problem under discussion is resumed on p. 13. 



DEAD RECKONING WITHOUT LOGARITHMS 11 



it affords a good opportunity of studying the principles 
underlying all mathematical tables in a case so simple as 
to offer no difficulty. 

MULTIPLICATION TABLE 
(to illustrate " argument" and " tabular number") 





2 


3 


4 


5 


6 


7 


8 


9 


10 


11 


12 


1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


11 


12 


2 


4 


6 


8 


10 


12 


14 


16 


18 


20 


22 


24 


3 


6 


9 


12 


15 


18 


21 


24 


27 


30 


33 


36 


4 


8 


12 


16 


20 


24 


28 


32 


36 


40 


44 


48 


5 


10 


15 


20 


25 


30 


35 


40 


45 


50 


55 


60 


6 


12 


18 


24 


30 


36 


42 


48* 


54 


60 


66 


72 


7 


14 


21 


28 


35 


42 


49 


56 


63 


70 


77 


84 


8 


16 


24 


32 


40 


48 


56 


64 


72 


80 


88 


96 


9 


18 


27 


36 


45 


54 


63 


72 


81 


90 


99 


108 


10 


20 


30 


40 


50 


60 


70 


80 


90 


100 


110 


120 


11 


22 


33 


44 


55 


66 


77 


88 


99 


110 


121 


132 


12 


24 


36 


48 


60 


72 


84 


96 


108 


120 


132 


144 



In this table the arguments are printed in heavy type and 
are contained in the left-hand column and the topmost 
horizontal line. In using the table, these arguments are 
given in pairs, being always the pair of numbers to be mul- 
tiplied. In fact, in the case of most tables, the arguments 
are thus given in pairs, though there are some tables with 
but a single argument. In the present case one number 
from the pair of arguments will be found in the left-hand 
column, the other in the top horizontal line. Thus, if we wish 
to multiply 6 and 8, these two numbers constitute the pair 
of arguments. We find the right line (belonging to 6) and 
column (belonging to 8), and the tabular number 48 (marked 
with a *) occurs at the intersection of the 6-line and the 8- 
column. If the pair of arguments are taken in the order 
8x6 instead of 6 X 8, we should use the 8-line and the 
6-column, again finding the required product (48) as the 
tabular number at the intersection. 



12 NAVIGATION 

Sometimes the given arguments cannot be found di- 
rectly in the table. Thus we might wish to multiply 
6| (written 6.5) by 8. Evidently the proper tabular 
number would be halfway between the 6x8 tabular 
number (48) and the 7x8 tabular number (56). The 
correct answer would therefore be 52. This process, by 
which the tabular number 52 is obtained, is called "in- 
terpolation." The example 6| X 8 is an extremely simple 
one. When less easy ones occur, the interpolation is best 
made as follows : we ascertain by subtraction how much 
the tabular number increases while the argument changes 
from 6 to 7. This increase is here 8, because the tabular 
number changes from 48 to 56 in the 8-column, while the 
argument in the left-hand column changes from 6 to 7. 
This increase of 8 in the tabular number is called a "tabular 
difference." We now compare the given argument (6.5) 
with the nearest argument (6) occurring in the left-hand 
column of arguments, and find an "argument difference" 
of 0.5 (being 6.5 minus 6). Since this "argument dif- 
ference" is 0.5, we must evidently take 0.5 X 8 (8 being the 
tabular difference), and increase the tabular number 48 by 
0.5 X 8, or 4. This again brings us to 52. Similar exam- 
ples are : 

(1) 5.3 X 4 = 21.2 ; (2) 7.7 X 8 = 61.6. 

In example (1) the tabular numbers are 20 and 24 ; the 
tabular difference is 4. 0.3 X 4 = 1.2; 20 + 1.2 = 21.2, the 
answer. Both examples may be verified, of course, by ordi- 
nary multiplication. 

When both given arguments contain fractions, as, for 
instance, 5.3 X 8.4, the resulting "double interpolation" 
is so complicated as to be of little practical use to the navi- 
gator. 

To make this general explanation of mathematical tables 
complete, it remains to show how they can be used in an 
inverse manner ; i.e. to find the argument from the tabular 



DEAD RECKONING WITHOUT LOGARITHMS 13 

number. Thus, if we were told that the tabular number is 
48, and one argument 8, an inspection of the table would 
at once show that the other argument must be 6. In this 
way the table might be used for division as well as multi- 
plication ; and interpolation would evidently also be possible. 
Many " mathematical tables must frequently be thus used 
in an inverse manner. 

Having thus explained the peculiarities of mathematical 
tables, we return to our dead-reckoning problem and its 
solution by means of the traverse table (p. 154). 

Referring to that table we find a column (p. 167), 
headed 40, the course angle of our present problem. On 
the left-hand side of the page we find the given distance, 63. 
Then, opposite the distance 63, and under 40, we find the 
latitude difference (abbreviated, "Lat.") and the departure 
(abbreviated, "Dep.") to be: 

lat. = 48.3, dep. = 40.5. 

The following are additional examples for practice : 

Given : dist., 84, course 26 ; Ans., lat. = 75.5, dep. = 36.8. 
Given : dist., 28, course 11 ; Ans., lat. = 27.5, dep. = 5.3. 

When the course is between 1 and 45 the course angle 
will be found in Table 1 at the head of the column : but when 
the course is between 45 and 90, it appears at the foot of 
the column. In the latter case, the tabular lat. and dep. 
are to be taken from the columns having "Lat." and "Dep." 
at the foot instead of the top of the column. Examples 
follow : 

Given : dist., 63, course 50 ; Ans., lat. = 40.5, dep. = 48.3. 
Given : dist., 84, course 64 ; Ans., lat. = 36.8, dep. = 75.5. 
Given : dist., 28, course 52 ; Ans., lat. = 17.2, dep. = 22.1, 

In addition to the course angles from 1 to 90, three ad- 
ditional angles are given in parentheses at the top and foot 
of each column. Thus, with the course angle 30 appear 
also 150, 210, 330. This simply means that the latitudes 



14 



NAVIGATION 




and departures are the same for these four 
course angles. The accompanying Fig. 5 shows, 
for instance, that the departures QP and Q'P' 
are equal for 30 and- 150 courses if the two 
distances YP and YP' are alike. 

It will be noticed also that our traverse table 
always gives distances from 1 to 50 on a left- 
hand page, and from 50 to 100 on a right-hand 
page. When distances larger than 100 occur, 
it is necessary to use the 100, 200, etc., given on 
the lower part of each page. If, for instance, 
we require the latitude and departure for a 
distance 363 miles, course 40, we turn again to 
the 40 column, and find (near the bottom of 
30 and 150. the page) : 

For 300 miles, lat. = 229.8, dep. = 192.8 
and (in the usual way) for 63 miles, lat. = 48.3, dep. = 40.5 
Sums, 363 =278.1 233.3 

Consequently, for dist. 363, course 40, lat. =278.1, dep. =233.3. 

Other examples are : 

Course 25, dist., 452 ; lat. = 409.6, dep. = 191.0. 
Course 68, dist., 521 ; lat. = 195.2, dep. = 483.1. 
Course 226, dist., 384 ; lat. = 266.8, dep. = 276.2. 

When the given distances or course angles, which are 
really the "pairs of arguments" (p. 11) of the traverse table, 
contain fractions, interpolation can be used ; but such close 
accuracy is seldom, if ever, required in navigation. 

More extended traverse tables will be found in Bowditch's 
"American Practical Navigator," published by the Navy 
Department, Washington. They are also printed separately 
in Bowditch's "Useful Tables." Both volumes can be 
purchased at any "navigation shop " where instruments 
and books suitable for navigators are sold. 

To complete this explanation of our traverse table, it is 
still necessary to mention that it also provides, with suf- 
ficiently close approximation, for the method of measuring 



DEAD RECKONING WITHOUT LOGARITHMS 15 

course angles in "points of the compass" (pp. 10, 41). This 
method is not now in use in the United States Navy, but it 
is still largely employed in merchant vessels. It is sufficient 
to state here that a course of 3 points, for instance, is very 
nearly equal to a course of 34, and the traverse table column 
for 34 may properly be used for a 3-point course. Similarly, 
31 may be used for 2f points, and the mariner desiring to use 
points can always find from the traverse table itself just 
what column to use. A special traverse table for points may 
also be found in Bowditch's Tables, already mentioned. 

We have now shown how to find latitude difference and 
departure by means of the traverse table. But our problem 
is not yet completely solved. Our ship (p. 8) started from 
the point Y in lat. 42 11' N. ; long. 59 28' W. She traveled 
63 miles on a 40 course, and the traverse table showed that 
she thus made good a latitude difference of 48.3 miles and a 
departure of 40.5 miles. It now remains to ascertain how 
much the ship changed her latitude in degrees and minutes 
from 42 11' N. and her longitude in degrees and minutes 
from 59 28' W. When we have found these last changes, 
we can learn the latitude and longitude of the point P, 
which we are required to find. 

To get the latitude change in degrees and minutes from 
the latitude difference in miles offers no difficulty. If the 
miles used are nautical miles (and in navigation they always 
are nautical miles), each mile of latitude difference corre- 
sponds to 1' of angular measure (p. 9), and 60 miles corre- 
spond to 1. Thus our ship must have changed her latitude 
48'.3, corresponding to a latitude difference of 48.3 miles. 
Her initial latitude having been 42 11' N., her final latitude 
at P will be 42 11' + 48' (if we omit the odd .3) or 42 59' N. 

The relation between departure and difference of longitude 
is not quite so simple. Our ship's departure of 40.5 miles 
might correspond to far more than 40.5 minutes of longitude. 
In fact, in very high latitudes near the north pole, the longi- 
tude meridians converge so closely that a person traveling 



16 NAVIGATION 

a few miles might change his longitude very greatly. At the 
pole itself a man might change his longitude 180 by simply 
stepping across the pole. So it follows that the longitude 
difference in minutes is greater than the departure in miles 
(however, cf . p. 4) . The difference between the two increases 
rapidly as we approach high latitudes though it is nil at 
the equator; in Table 2 (beginning p. 168) we give this 
excess of longitude difference over departure for all latitudes 
under 60, and for all longitude differences up to 100. When 
the longitude differences are greater than 100, it is necessary 
to use the numbers given for 100, 200, 300, etc., near the 
bottom of each page in the table, and to sum tabular num- 
bers, precisely as we did with the traverse table. 

It will be noticed that Table 2 gives "tabular numbers" 
for each degree of latitude in a separate column, and that 
these various latitudes are called "middle latitudes." Thus 
the middle latitude and the longitude difference are the pair 
of arguments (p. 11) for Table 2, and, as we shall see pres- 
ently, the use of the middle latitude avoids any uncertainty 
in choosing the correct column for use. In our present 
problem we have at our disposal (p. 15) two different lat- 
itudes : the initial latitude at the point F, 42 11' N., and 
the final latitude at the point P, 42 59' N. In this case, the 
two latitudes are so nearly equal that we might use either 
of them as an argument in Table 2 without material inaccu- 
racy. In fact, in using Table 2 it is unnecessary to consider 
minutes of latitude, the nearest degree being sufficient. 

But often the two latitudes available at this stage of the 
problem differ by many degrees. In such cases mariners 
always use the average of the two latitudes, and call it the 
"middle latitude." In the present case, the middle latitude 
would be found thus : 

Initial latitude = 42 11' 
Final latitude = 42 59' 
Sum = 85 10' 
\ sum = middle latitude = 42 35' 



DEAD RECKONING WITHOUT LOGARITHMS 17 

The nearest even degree to 42 35' is 43, and the prob- 
lem would therefore be worked with the 43 column of middle 
latitude in Table 2. 

Before completing our problem it is necessary to point 
out that while Table 2 is intended primarily for changing 
longitude differences in minutes into departures in miles, it 
can also be used (as stated at the foot of each page) for the 
inverse transformation of departures into longitude dif- 
ferences ; and this is the transformation we must make in 
our present problem. It is merely necessary to use the 
departure (40.5) in the left-hand column, at the head of 
which are the words "Long. Diff. or Dep.," indicating that 
either of these two may be used as the argument in that 
column. Then, in the 43 column of middle latitude, we 
find (using interpolation) the tabular number 10.8. 

This means that a longitude difference of 40'. 5 corre- 
sponds to a departure of 40.5 10.8 miles, or 29.7 miles. 

But when the table, as in the present case, is used for the 
inverse transformation, the tabular number 10.8 must, 
before use, be multiplied by the factor given at the bottom 
of the column. For the middle latitude 43 this factor is 
1.37; and so the right tabular number becomes, in the 
present case : 

10.8 X 1.37 = 14.8; 

and as the longitude difference is always greater than the 
departure, it follows that the departure of 40.5 miles gives 
a longitude difference of : 

40.5 + 14.8 = 55'.3 = 55', 

if we omit the odd tenths. 

The initial longitude of the ship at the point Y was 
59 28' W. As her 40 course has carried her nearer to Green- 
wich, it follows that her final longitude at the point P is : 

59 28' W. - 55' = 58 33' W. 
We shall now discuss the following similar problem : 
A ship takes her departure from a point about one mile 
c 




18 NAVIGATION 

east of Navesink Highlands Light, New Jersey, in the initial 
lat. 40 24' N., initial long. 73 58' W., and travels 1377 
miles on a course of 166. What final latitude and longitude 
does she attain ? 

Entering the traverse table in the column headed 166, 
which is the same as the 14 column, we find : 

For dist. 900, lat., 873.2, dep., 217.7 

For dist. 400, lat., 388.1, dep., 96.7 

For dist. 77, lat., 74.7, dep., 18.6 

Sums, 1377, 1336.0, 333.0 

To make the large given distance (1377 miles) come within 
the range of Table 1, it has been necessary to enter the 166 
column three times, with the arguments 900, 400, and 77, 
and then to sum the corresponding tabular numbers. 

The latitude difference, 1336 miles, is equivalent to 1336', 
or 22 16', counting, as usual, 60' to 1. Then, since the 
direction of her course (166) carried the ship to the south 
of her initial position (cf. Fig. 5, p. 14, and p. 19), we have : 

Initial lat., 40 24' N. 
Lat. diff., 22 16' N. 
Final lat., 18 8' N. 
Middle lat., 29 16' N. 

Now turning to Table 2, hi the proper column for middle 
latitude 29 : 

For dep. 300 tabular number is 37.6 
For dep. 33 tabular number is 4.1 
Sums 333 41.7 

As in the former example, this 41.7 must be multiplied 
by the factor at the bottom of the column. This factor is 
1.14. Multiplying, we have: 41.7 X 1.14 = 47.5. Conse- 
quently, long. diff. = 333 + 47.5 = 380'.5 = 6 20'.5. Since 
the direction of her course (166) carried the ship eastward, 
and therefore nearer to Greenwich, it follows that her final 
longitude is 73 58' W. - 6 20', or 67 38' W. The final 
position is therefore : lat. 18 8' N. ; long. 67 38' W. 



DEAD RECKONING WITHOUT LOGARITHMS 19 

' The point indicated by this final latitude and longitude 
is just off the entrance to the Mona Passage, between Haiti 
and Porto Rico ; the given course and distance would there- 
fore be correct for a voyage from New York to Mona Passage 

Additional similar problems are : 

1. Initial lat., 40 28' N. ; initial long., 73 50' W. ; course, 
119; dist., 2924 miles. This would take the ship from 
Sandy Hook to St. Vincent, Cape Verde Islands. 

Ans. Final lat., 16 50' N. ; final long., 25 7' W. 

2. Initial lat., 40 10' N. ; initial long., 70 0' W. ; course, 
75 ; dist., 2606 miles. This would take the ship from Nan- 
tucket Lightship to Fastnet, the nearest point of the Irish 
coast. 

Ans. Final lat., 51 24' N. ; final long., 9 37' W. 

Before proceeding to our second fundamental problem 
(p. 8), it will be well to explain briefly two further points 
of interest. The first of these relates to the method of desig- 
nating a ship's course. We have hitherto supposed it to 
be measured in degrees, from the north, around by way of 
the east, through the south and west, and so back to the 
north again. This is the best way to count courses, and is 
the way now in use in the United States Navy. Since a 
whole circle contains 360, it follows that courses may con- 
tain any number of degrees from to 360. 

But there is another quite convenient, although older, way 
of designating courses, in which a 60 course, for instance, is 
written N. 60 E., showing that the ship must be steered 60 
east of north. In a similar way, a 120 course is written 
S. 60 E., showing that the helmsman should head her 60 
east of south, which would be the same as 30 south of east, 
or 120 from the north toward the south by way of east. 

The second further point of interest has to do with the 
relation between Tables 1 and 2. It is possible to avoid 
entirely the use of Table 2, and to transform longitude differ- 
ences into departures, and vice versa, by means of Table 1 



20 NAVIGATION 

alone. It so happens that the relation between these two, 
for any given middle latitude, as, for instance, 29, is iden- 
tical with the relation between distance and latitude difference 
in Table 1 for the course 29. In other words, if we have 
given a middle latitude and a longitude difference, and wish 
to find the departure, we : 

Call the middle latitude a course, and 
Call the longitude difference a distance ; 

Then, corresponding to that course and distance, find from 
Table 1 the tabular latitude difference, and it will be 
the required departure. The same process can also be 
reversed, so as to find the longitude difference from the 
departure. 

While this method with Table 1 is quite correct, we believe 
beginners (at least) will find the use of Table 2 advantageous 
in the solution of these problems, especially when the middle 
latitude is not very great. 

Coming now to our second fundamental problem of dead 
reckoning, let us suppose a ship is required to proceed from 
the initial lat. 42 11' N. and long. 59 28' W. to a final 
lat. 42 59' N. and long. 58 33' W. We are to find the course 
she must steer, and the distance she must run. 

We have at once the latitude difference of 48', or 48 miles, 
and the middle latitude 42 35', or nearest whole degree of mid- 
dle latitude, 43. The longitude difference is 55' ; and with this 
we find from Table 2 the correction 14.8 in the 43 column 
of middle latitude. Remembering that this time we are 
transforming a longitude difference into departure, and con- 
sequently do not need to use the factor at the foot of the 
column, we subtract this correction (14.8) from the longi- 
tude difference (55') and obtain the departure as 40.2 miles. 

Next we proceed to Table 1, to find the course and distance 
corresponding to lat. 48, dep. 40.2. To do this, we must 
find a place in Table 1 where this particular latitude and 
departure appear side by side. If this pair of numbers 



DEAD RECKONING WITHOUT LOGARITHMS 21 

cannot be found (exactly) side by side, we must take the 
pair which come nearest to them : in this case such a pair 
of numbers is found in the 40 course column, opposite dist. 
63. So it appears that the ship must steer on a 40 course 
a distance of 63 miles, to proceed from the given initial to 
the given final latitude and longitude. This problem is the 
direct converse of the one first solved (pp. 15, 17). 

As a second example, let us now calculate the course and 
distance from Sandy Hook, lat. 40 28' N. ; long. 73 50' W., 
to St. Vincent, lat. 16 50' N. ; long. 25 7' W. We have, 
by subtraction, lat. diff. = 23 38' = 1418' = 1418 miles; 
long. diff. = 48 43' = 2923'. 

This 2923' must be turned into a departure, the middle 
latitude being 28 39', or, to the nearest whole degree, 29. 
Turning to the column of Table 2 which belongs to 29 of 
middle latitude, we find the correction for 2923' of longitude 
difference thus : 

Tabular number for 900 = 113.0, 

which being multiplied by 3, gives : 

Tabular number for 2700 = 339.0 

Also, tabular number for 200 = 25.1 

Tabular number for 23 = 2.9 

Sums, tabular number for 2923 = 367.0 

This must be subtracted from the longitude difference, and 

so we get : 

dep. = 2923 - 367.0 = 2556 miles. 

We have now to seek a place in Table 1 where lat. 1418 and 
dep. 2556 appear side by side. No traverse tables are suffi- 
ciently extended to contain these large numbers, but we 
can at once obtain an approximate answer to the problem 
by dividing both numbers by 100. This reduces them to 
lat. 14.2, dep. 25.6 ; and the nearest numbers to these which 
can be found side by side in Table 1 are in the column belong- 
ing to course 119 and opposite dist. 29. This course (119) 
is the same as would have been obtained if we had not been 



22 NAVIGATION 

forced to divide our latitude and departure by 100, to bring 
them within the range of Table 1. But the dist. 29 must 
now be multiplied by 100, to remove the effect of our former 
division of latitude and departure by 100. Thus we have 
the closely approximate information that the course and 
distance from Sandy Hook to St. Vincent are 119 and 2900 
miles. The same problem (p. 19), when taken in its inverse 
form, starts with the numbers 119 and 2924 miles. 

In discussing such a problem, many beginners have dif- 
ficulty in choosing correctly the course number (119) from 
the four (61, 119, 241, 299) to be found at the foot of 
the same column of Table 1. This choice is easily made with 
the help of our knowledge of elementary geography, or with 
any rough chart or map. From these, we know that St. 
Vincent is south and east of Sandy Hook, and the only one 
of the four possible courses that will carry a ship south and 
east is course 119. The same course might be written in 
the other notation (p. 19) S. 61 E., which possibly makes 
the actual direction to be steered a little easier to under- 
stand. 

The above result is approximate only, but higher accuracy 
is seldom required. When desired, it can be obtained by 
certain kinds of interpolations (p. 12) ; but these are always 
unsatisfactory, especially as complete precision can always 
be easily had by the use of logarithms, as explained in the 
next chapter. 



CHAPTER III 

DEAD RECKONING WITH LOGARITHMS 

SINCE the publication in 1876 of Kelvin's tables for 
facilitating Sumner's method, it has been possible to navi- 
gate in the most approved way without using logarithms or 
trigonometry. Those who desire to study the subject in 
this manner may do so by simply omitting those parts of 
the book in which logarithmic or trigonometric formulas 
and calculations occur. But this method of study is not 
recommended, except perhaps for a first reading; for a 
knowledge of logarithmic processes always affords a most 
desirable check on the accuracy of the other method, and 
so makes for safety of the ship and peace of mind of the 
navigator. 

Proceeding, then, with the subject of logarithms, we may 
define them as a mathematical device for facilitating calcula- 
tions. They are merely numbers; but they are numbers 
having this peculiarity : every logarithmic number belongs 
to some ordinary number (like 1, 2, 3, 27, 800, etc.), and 
belongs to it, alone. Its logarithm belongs to the number as 
a man's shadow belongs to the man. 

For our present purpose it is unnecessary to enter into the 
theory of logarithms ; we shall explain only the methods of 
using them in practice. Logarithms (abbreviated "log") 
always consist of two parts, a "whole number" part and a 
"decimal" part. Thus, 3.30103 is a logarithm, of which 
the whole number part is 3, and the decimal part .30103. 
The whole number part may even be zero : thus, 0.30103 
is also a logarithm. The decimal part of the logarithm 
is found from a table of logarithms, such as our Table 3 

23 



24 NAVIGATION 

(p. 178) ; but the whole number part is found by an inspec- 
tion of the number to which the logarithm belongs. 

We shall hereafter, to save space, always write "log 26" 
in place of "the logarithm belonging to 26": and, with 
the help of this abbreviation, we may now write the follow- 
ing tabular statement, which is fundamental in the matter 
of logarithms : 

log 1 = 0.00000, log 1000 = 3.00000, 
log 10 = 1.00000, log 10000 = 4.00000, 
log 100 = 2.00000, log 100000 = 5.00000, etc. 

In other words, for these particular numbers, all "mul- 
tiples" of 10, the decimal part of the log is zero. For 
numbers intermediate between 1 and 10, the whole number 
part of the log is 0, and the decimal part lies between 
.00000 and .99999. For those between 10 and 100 the whole 
number part is 1, and the decimal part again lies between 
.00000 and .99999. 

The general rule is : the whole number part of a log is 
one less than the number of figures or "digits" in the number 
to which the log belongs. Thus, the number 26 has two 
digits : the whole number part of its log is 1. The number 
2678 has four digits : the whole number part of its log is 
therefore 3. 

If a number is itself partly decimal, we count only the 
number of digits to the left of the decimal point for the pur- 
poses of the present rule. Thus, 26.78 has two digits only; 
2.678 has 'one; 267.8 has three, etc. 

If, on the other hand, a number is wholly decimal, as 
0.2678, the whole number part of its logarithm should be 
"negative," or minus, i.e. less than 0; and it will be one 
greater than the number of zeros immediately following the 
decimal point in the number. According to this, the whole 
number part of log 0.2678 should be 1, because this 
number has no zeros immediately following the decimal 
point. But as these negative whole number parts are 
very inconvenient in actual work, it is customary to increase 



DEAD RECKONING WITH LOGARITHMS 25 

all logs of decimal numbers arbitrarily by 10, which will 
avoid the negative sign. This arbitrary increase is always 
corrected again in the further or final procedure, so that it 
cannot possibly introduce error into the work. 

In the case of log 0.2678, the arbitrary increase of 10 
changes the 1 to + 9 l ; and so 9 would be the whole 
number part of log 0.2678. Similarly, log 0.002678 would 
have 7 for its whole number part, because there are two zeros 
after the decimal point. This would make the whole number 
part of the log 3, which, being increased by 10, gives + 7. 

In general, this matter of logs of wholly decimal numbers 
may be summarized as follows : 

log 0.1 =9.00000, log 0.0001 =6.00000, 
log 0.01 =8.00000, log 0.00001 =5.00000, 
log 0.001 = 7.00000, log 0.000001 = 4.00000, etc. 

In all these cases the decimal part of the log is zero : 
and if the number lies, for instance, between 0.1 and 0.01, 
the whole number part of the log will be 8, and the decimal 
part will lie between .00000 and .99999. 

The decimal part in the log of any number is taken from 
Table 3 without regard to the position of the decimal 
point in the number itself. The numbers 0.2678, 0.002678, 
26.78, 2.678, 267.8, and 2678 all have precisely the same 
decimal part in their logs, so that such logs will differ in 
their whole number parts only. We can at once obtain this 
common decimal part from Table 3 (p. 181), where it is 
found to be .42781. In looking up this log, we again use 
(p. 11) a pair of arguments. The argument for the left- 
hand column consists of the first three digits of 2678 (267) ; 
and in selecting this argument we disregard any zeros that 
may immediately follow the decimal point, if the number 
is wholly decimal, like .002678. The other argument, in 
the top horizontal line of the tabular page is 8, the right- 
hand digit of the number 2678. In the horizontal line 

1 According to Algebra, 9 is greater than - 1 by 10. 



26 NAVIGATION 

opposite 267, and in the column headed 8, appears 781 ; and 
these are the last three digits of the required log (.42781). 
The first two digits (.42) are common to a great many logs, 
and are therefore only printed in the column headed 0. 
The first two digits of every log are thus taken from the 
zero column, regularly from the same horizontal line that 
contains the last three digits of the log, or from some line 
above it. Only when there is an asterisk printed in the table 
with the last three digits do we make an exception, and take 
the first two digits from tha line below the one containing the 
last three. Thus the decimal part of log 2691 is .42991, but 
the decimal part of log 2692 is .43008. 

Having thus found the decimal part of log 2678 to be 
.42781, and the number 2678 having four digits, the com- 
plete 

log 2678 = 3.42781 ; 

and here the reader should once more note that all tabular 
logs like .42781 are thus always decimals. The correspond- 
ing logs for the other numbers given above are : 

log 267.8 = 2.42781, 
log 26.78 = 1.42781, 
log 2.678 = 0.42781, 
log 0.2678 = 9.42781, 
log 0.002678 = 7.42781. 

It is clear that Table 3 gives directly the decimal part of 
the logs of all numbers containing four digits. If the number 
contains less than four digits, as 26, we should look it up in 
the table as if it were 2600. We should find 260 as the 
argument in the left-hand column (p. 181) ; and in the 
corresponding line, in the column headed (the fourth digit 
of 2600), is 41497. This is the decimal part, as usual, and 
the complete 

log 26 = 1.41497. 

If, on the other hand, the number whose log is wanted 
contains more than four digits, as 26782, it is necessary to 



DEAD RECKONING WITH LOGARITHMS 27 

resort to interpolation (p. 12). The number of digits being 
here 5, the whole number part of the log is 4 (p. 24). The 
decimal part of the log is to be found quite without regard 
to decimal points (p. 25). It may therefore be taken 
from Table 3 just as if we wanted log 2678.2 instead of 26782. 
Now the table tells us (p. 181) : 

decimal part of log 2678 = 42781, 
decimal part of log 2679 = 42797. 

The tabular difference (p. 12) of these two decimal parts 
is 16. As 26782 may, for our present purpose, be regarded 
as lying & of the way from 2678 to 2679, it follows that the 
decimal part of log 26782 will lie ^ of the way from 42781 
to 42797. Evidently, we must multiply the tabular differ- 
ence 16 by -$ (giving 3.2) to find how much larger the decimal 
part of log 26782 is than the decimal part of log 2678. 
This 3.2 (or 3, in round numbers) must then be added to 
42781 ; and we have, as the result of this interpolation : 

decimal part of log 26782 = .42784. 

As we have just found the whole number part to be 4, 
we have for the complete : 

log 26782 = 4.42784. 

This whole process of interpolation may perhaps be more 
clearly understood if we repeat (p. 10) that all tables furnish 
tabular numbers corresponding to given arguments. In- 
terpolation is necessary when the given arguments are not 
to be found in the argument part of the table, but fall 
between two of the tabular arguments. Then we obtain 
by subtraction the difference between the given argument 
and the nearest smaller argument contained in the table. 
This difference is the "argument difference" (abbreviated, 
arg. diff.), and it should be expressed as a decimal fraction 
of the interval between two successive arguments (cf. $, 
above). The tabular difference (tab. diff.) between two 
successive tabular numbers being also obtained by subtrac- 



28 NAVIGATION 

tion, we have only to multiply the tabular difference by the 
argument difference to find the "interpolation difference" 
(int. diff.)- This is then added 1 to the proper tabular 
number (belonging to the above-mentioned nearest argu- 
ment given in the table) to obtain the tabular number re- 
quired. 

The multiplication of the tabular difference by the argu- 
ment difference is facilitated by certain little auxiliary mul- 
tiplication tables (called tables of "proportional parts") 
printed in the margins of many mathematical tables. In 
the example given above, the tabular difference was 16 ; and 
Table 3 contains on the proper page (p. 181) a proportional 
part table headed with this same number 16 ; and it shows 
that for an argument difference .2, and tabular difference 16, 
the interpolation difference is 3.2, just as we found above. 

Other examples of logarithms are : 

log 427 = 2.63043, log 42765 = 4.63109, 

log 4276 = 3.63104, log 282374 = 5.45082, 

log 0.4276 = 9.63104, log 2 = 0.30103, 

log 0.42765 = 9.63109, log .0027 = 7.43136. 

The above considerations are preparatory only to the 
actual use of Table 3 ; and they are not yet quite complete. 
For it is still necessary to explain the inverse use (p. 12) of 
the table, or, in other words, the finding of the number to 
which a given log belongs. Thus, if the given log were 
3.42781, we should begin by looking up its decimal part 
among the logs in the table. Finding it there, we take out 
the number to which it belongs, 2678. We then put in the 
decimal point according to the whole number part of the log. 
This being 3, we know (p. 24) that the number required must 
contain 4 digits. Therefore : 

number to which the log 3.42781 belongs = 2678. 

1 Except when a glance at the table shows that the tabular num- 
bers are growing smaller, in which case the interpolation difference 
must be subtracted. This never occurs in Table 3, but happens fre- 
quently in Table 4. 



DEAD RECKONING WITH LOGARITHMS 29 

If the given log had been 2.42781, the table would furnish 
the same number 2678, but the decimal point would be 
differently located. Because the whole number part of the 
given log is now 2, we know that the number to which it 
belongs has three digits, and so : 

number to which the log 2.42781 belongs = 267.8. 

When the given log is not to be found in the table exactly, 
a process of inverse interpolation is, of course, necessary. 
Thus, if the given log is 4.42784, we look for its decimal 
part in the table, and find it lies between 

42781, which belongs to the number 2678, and 
42797, which belongs to the number 2679. 

The decimal part of the given log being 42784 is greater by 
3 than the nearest tabular number 42781. This 3 is there- 
fore the interpolation difference. The tabular difference is 
16, obtained by subtraction between 42781 and 42797. We 
now divide the interpolation difference by the tabular dif- 
ference, which gives .2 (^ = 0.2, in round numbers). This 
.2 is the argument difference, and therefore the complete 
number belonging to the decimal part of 'the log (42784) 
is 26782. The whole number part of the given log 
being 4, the required number must have 5 digits, and will 
therefore be 26782. Had the given log been 2.42784, we 
should have arrived at the number 26782 in just the same 
way; but we should locate the decimal point differently. 
The whole number part of the log being now 2, there should 
be only 3 digits in the number, and we should have : 
number to which the log 2.42784 belongs = 267.82. 

Other similar examples are : 

log = 2.71828, corresponding number = 522.73, 
log = 4.26323, corresponding number = 18333, 
log = 9.26323, corresponding number = 0.18333, 
log = 0.21000, corresponding number = 1.6218. 

The reader will perceive, from a consideration of these 
interpolated numbers, that work with logarithms is never 



30 NAVIGATION 

exact, absolutely. This is inherent hi the nature of our 
log tables, which really contain only the decimal parts of the 
logs carried out to five places of decimals. Further decimals 
of course exist, but are here omitted, because five places 
always give sufficient accuracy for navigation calculations. 

The simplest calculations which are facilitated by loga- 
rithms are the ordinary arithmetical processes of multi- 
plication and division. These processes can be turned into 
addition and subtraction by the use of the following 
principle : 

The log of a product is equal to the sum of the logs of the 
factors. 

According to this principle, if we wish to multiply a series 
of factors, we simply add their logs. The sum is then a log 
and the number to which this log belongs is the product of the 
series of factors. Suppose, for instance, we wish to multiply 
the factors 2, 3, and 4. The product should be 24. Proceed- 
ing with logs, we have from Table 3 : 

log 2 = 0.30103, 

log 3 = 0.47712, 

log 4 = 0.60206, 

log product = sum = 1.38021, 

and the number to which the log. 1.38021 belongs is, accord- 
ing to Table 3, 24.00, the correct product. 

It is evident that the use of the log table is here of no 
advantage, because the factors are very small : but when 
large numbers are to be multiplied the advantage is very 
great. 

Taking now a similar simple example of division, let us 
divide 6 by 3. In division, evidently, we must subtract 
the log of the divisor from the log of the dividend, to obtain 
the log of the quotient. We have 

log 6 = 0.77815, 

log 3 = 0.47712, 

log | = difference = 0.30103, 



DEAD RECKONING WITH LOGARITHMS 31 

and the number to which the log 0.30103 belongs is 2.000, 
the correct quotient. Other examples are : 

2.426 X 42.78 X 17.26 = 1791 .3, 

6.242 X 87.24 x 62.71 = 34149, 

ff|= 1.6234, 

24 = ' 75 ' 

In the last example, we have 

log 18 = 1.25527, 
log 24 = 1.38021. 

The subtraction would lead to a negative log because 1.38021 
is larger than 1.25527. Therefore we arbitrarily increase 
1.25527 by 10, giving 11.25527, and then the subtraction 

gives 

log quotient = 9.87506, 

which is the log belonging to the number 0.75, the correct 
quotient. 

We come now to the solution of the two fundamental 
problems of dead reckoning (pp. 8, 10) by means of logs. 
For this purpose we must use our Table 4, in connection with 
Table 3. Table 4 is called a trigonometric log table and 
the tabular numbers in it are certain logs known as : 
sine, abbreviated sin, cotangent, abbreviated cot, 
cosine, abbreviated cos, secant, abbreviated sec, 
tangent, abbreviated tan, cosecant, abbreviated esc. 

It is not our purpose to consider the theory of trigonom- 
etry, but it is necessary for the reader to have 
some understanding of its practical applica- 
tions. If we have a triangle QPY (fig. 6), we 
notice that it is made up of six "parts," the 
three sides and the three angles. Now it is a 
fact that if we know any three of these six y 
parts, we can calculate the other three parts, FIG. 6. Trigo- 
provided one of the known parts is a side. 
Trigonometry is the branch of mathematics which enables us 



32 NAVIGATION 

to do this, and the triangle QPY is the very triangle which 
occurs in the two problems of dead reckoning, 

In trigonometry, every angle has belonging to it a sin, 
cos, etc., just as every number has its log. These sines, 
etc., can be taken out of Table 4 by means of a pair of argu- 
ments in the usual way. The two arguments are the number 
of degrees and the number of minutes in the angle (p. 9). 
The number of degrees is found in Table 4 at the top or bottom 
of the page, and the number of minutes in the right-hand or 
left-hand column. Each page (as, for instance, p. 229) has 
eight degree numbers, four, 33, (213), (326), and 146 at 
the top, and four, 123, (303), (236), and 56 at the bottom. 
The proper sines, etc., for all these degrees appear on the 
same page (p. 229). When the degree number is at the top 
or bottom of the left-hand column 33, (213), (303), and 
123, the minutes must be taken from the left-hand column. 
But when the number of degrees is at the top or bottom of the 
right-hand column 146, (326), (236), and 56, the minutes 
must come from the right-hand column. And when the 
number of degrees comes from the top of the page, we must 
look for the proper sine, etc., in a column having the word 
sin, etc., at the top. But when the degree number comes 
from the bottom of the page, the sine, etc., will be taken 
from a column having the word sin, etc., at the bottom. 
Thus (p. 229) : 

sin 33 26' = sin 146 34' = cos 56 34' = cos 123 26' = 9.74113. 

In this way, sines, tangents, etc., can be taken from 
Table 4. Examples are : 

sin 28 32' = 9.67913, cot 117 10' = 9.71028, 
cos 66 14' = 9.60532, sec 12 40' = 0.01070, 
tan 128 28' = 0.09991, esc 111 11' = 0.03038. 

These sines, etc., are really all logs. When the whole num- 
ber part is 9, it indicates that the log belongs to a number 
which is wholly decimal (see p. 24), and that the log has 
been arbitrarily increased by 10. 



DEAD RECKONING WITH LOGARITHMS 33 

Of course these trigonometric tables can also be used in 
the inverse manner. Thus, to find the angle corresponding 
to the sin 9.28190, we turn to p. 207, and finding 9.28190 in 
the sin column, we see that the corresponding angle is 
either 11 2', 191 2', 168 58', or 348 58'. When the sin, 
etc., cannot be found in the table exactly, we may always 
take the nearest one : interpolation is never practically 
necessary in using the trigonometric tables in navigation. 
Examples are : 

sec = 0.17177, angle = 47 40', 227 40', 132 20', or 312 20', 
tan = 0.17177, angle = 56 3', 236 3', 123 57', or 303 57', 
sin = 9.17177, angle = 8 32', 188 32', 171 28', or 351 28', 
cos = 9.17177, angle = 81 28', 261 28', 98 32', or 278 32', 
esc = 0.17177, angle = 42 20', 222 20', 137 40', or 317 40', 
cot = 0.17177, angle = 33 57', 213 57', 146 3'i or 326 3'. 

Having thus explained the use of Table 4, we shall now 
apply it to the two problems of dead reckoning. These 
problems are : 

1. To find latitude difference and departure from course 
and distance ; 

2. To find course and distance from latitude difference 
and departure. 

These problems are solved by means of the following 
formulas, in which the letter C represents the course angle : 

n . f log lat. diff. = log dist. + cos C, 
" J [ log dep. = log dist. + sin C. 

I tan C = log dep. log lat. diff., 

* j log dist. = log dep. sin C. 

Sometimes it is preferable to find the distance from the 
latitude difference instead of the departure. We then use 
the following modification of formula (2) : 

(2') log dist. = log lat. diff. - cos C. 

Let us now solve with these formulas our former problem 
(p. 18), in which a ship traveled 1377 miles on a course of 
166. Applying formula (1) above, we have : 



34 NAVIGATION 

log dist. (1377) =3.13893 log dist. (1377) =3.13893 

cos C (166) = 9.98690 sin C (166) = 9.38368 

sum = log lat. diff. = 3.12583 x sum = log dep. = 2.52261 1 

corresponding lat. diff. = 1336.1 corresponding dep. = 333.1 

These corresponding latitude difference and departure 
agree very closely with the results already found (p. 18) 
from Table 1. 

If the departure and latitude difference were given, we 
could find the course and distance by means of formula (2)- 
In the present case we have : 

log dep. (333.1) =2.52261 log dep. (333.1) =2.52261 

log lat. diff. (1336.1) = 3.12583 sin C (166) = 9.38368 

by subtraction, tan C = 9.3967S 2 by subtraction, log dist. = 3.13893 3 

corresponding C = 166 corresponding dist. = 1377 

These numbers, 166 and 1377 miles, are the same numbers 
with which we began this calculation ; so it is clear that the 
log method of calculation agrees with the traverse table 
method. For accuracy the log method is superior. 

The transformations of departure into longitude differ- 
ence, and vice versa, are accomplished logarithmically with 
the following formulas : 

(3) log long. diff. = log dep. cos middle lat. 

(4) log dep. = log long. diff. + cos middle lat. 

Thus the longitude difference corresponding to dep. 333.1 
would be calculated by formula (3) as follows : 

log dep. (333.1) =2.52261 

cos mid. lat. (29 16'; p. 18) = 9.94069 
by subtraction, log long. diff. = 2.58192 
corresponding long. diff. = 381 '.9 = 6 21 '.9. 

1 These numbers have been diminished by 10, to allow for the fact 
that both cos C and sin C have been arbitrarily increased by 10 (p. 
32; cf. also p. 25). 

2 This number has been increased by 10, and therefore is in accord 
with the usual practice of avoiding negative whole numbers in the 
trigonometric Table 4. 

3 This subtraction is correct, if we remember that the 9.38368 is 
really too large by 10. 



DEAD RECKONING WITH LOGARITHMS 35 

This is in close accord with the result on p. 18, where 
Table 2 gave 6 20'. 5. The logarithmic method is again 
the more precise, for it takes account of minutes in the course, 
which were neglected on p. 18. But either result is accurate 
enough for practical purposes. 

Before finally leaving these problems of dead reckoning, 
we shall explain briefly two additional methods of solving 
them which differ from the method so far employed. These 
two additional methods are called "Mercator sailing" and 
"great circle sailing"; whereas, up to the present, we have 
been using "middle latitude sailing," so named because 
the middle latitude appears in the calculations. 

Mercator sailing is based on a kind of chart first designed 
by Gerhard Mercator, a sixteenth century geographer. 
Such charts are still widely used for nautical purposes. 
In calculations based on them, every parallel of latitude is 
referred directly to the equator by means of a table of "merid- 
ional parts." Our Table 5 is such a table, and it gives the 
meridional part for every degree and minute of latitude 
from the equator to 60. These meridional parts are really 
the distances from the equator to the several parallels of 
latitude, such as they would appear on a Mercator chart 
drawn to such a scale that 1' of longitude at the equator would 
occupy one linear unit on the chart. Thus the meridional 
part for lat. 40 is given in Table 5 as 2607.6. Suppose the 
scale of the chart at the equator were 1 inch to the degree of 
longitude. That would be $ inch to the minute. The dis- 
tance on the chart from the equator to the 40 parallel of 
latitude would then be 2607.6 X ^ inches = 43.46 inches. 
It is needless to say that a chart on such a scale could not 
show a very large part of the ocean on a single sheet. 

Calculations by Mercator sailing are of course only made 
when the distances involved are large and great accuracy is 
required. It is therefore best to do them by means of 
logarithms, although it is also possible to obtain Mercator 
results from the traverse table . In such calculations we do not 



36 NAVIGATION 

use the latitude difference of ordinary middle latitude sailing. 
In its place appears the "meridional latitude difference" (ab- 
breviated mer. lat. diff .), defined as the difference between the 
meridional parts (Table 5) belonging to the two latitudes 
(initial and final) involved in the problem. With this defini- 
tion in mind we may now give the Mercator formulas as 
follows : 

(5) log mer. lat. diff. = log long. diff. + cot C. 

(6) log long. diff. = log mer. lat. diff. + tan C. 

(7) tan C = log long. diff. - log mer. lat. diff. 

Let us now apply these formulas to the problem of pp. 18 
and 33, in which a ship starts from the initial lat. 40 24' N. ; 
long. 73 58' W., and travels 1377 miles on a course, C, 
of 166. What final latitude and longitude does she at- 
tain ? The latitude difference is found in the ordinary way 
(p. 34), there being no special Mercator formula for it, and 
comes out 1336.1 miles, or 1336M = 22 16'. The final lati- 
tude (p. 18) is therefore 40 24' - 22 16' = 18 8'. Then, 
from Table 5, we have : 

for initial lat. 40 24', mer. parts = 2638.9 
for final lat. 18 8', mer. parts = 1099.4 
by subtraction, 1 mer. lat. diff. = 1539.5 

Now, applying formula (6), we have: 

log mer. lat. diff. (1539.5) (Table 3, p. 179) = 3.18738 
tan C (166) (Table 4, p. 209) = 9.39677 

by addition, log long. diff. = 2.58415 

corresponding long. diff. (Table 3, p. 183) = 383'.8 = 6 24' 

The final longitude is therefore 73 58' - 6 24' = 67 34' W., 
whereas we obtained before 67 38' W. (p. 18). 

Finally, we shall apply the Mercator method to the 
example of p. 21. It is required to find the course and 
distance from 

Sandy Hook, lat. 40 28' N. ; long. 73 50' W. to 
St. Vincent, lat. 16 50' N. ; long. 25 7' W. 

1 If one latitude were in the southern hemisphere and the other 
in the northern, we should add the meridional parts. 



DEAD RECKONING WITH LOGARITHMS 37 

We have from Table 5 : 

for initial lat. 40 28', mer. parts = 2644.2 
for final lat. 16 P 50', mer. parts = 1018.1 
by subtraction, mer. lat. diff. = 1626.1 

The longitude difference is found by subtraction to be 
73 50' - 25 T = 48 43' = 2923'. Now applying formula 
(7), we have : 

log long. diff. (2923) (Table 3) = 3.46583 
log mer. lat. diff. (1626) (Table 3)= 3.21112 
by subtraction, tan C = 0.25471 

and therefore (Table 4) C = 119 5'. 

The distance is found in the ordinary way from the 

latitude difference (not mer. lat. diff.) by means of formula 

(20, P. 33. 
The latitude difference is 40 28'- 16 50' = 23 38' = 1418'. 

Formula (2') then gives : 

log lat. diff. (1418') (Table 3) = 3.15168 
cos C (119 5') (Table 4) = 9.68671 1 

by subtraction, log dist. = 3.46497 1 

corresponding dist. (Table 3) = 2917 

Course 119 5', distance 2917 miles is therefore the 
solution by Mercator sailing. On p. 22, we obtained 119 
and 2900 miles; and on p. 19 we began with 119 and 2924 
miles. The agreement is satisfactory. 

Having thus briefly described Mercator sailing, we come 
next to "great circle sailing." This is a method of determin- 
ing the ship's course toward her port of destination in such a 
way that the distance to be traveled will be as short as 
possible. If the earth's surface were flat instead of spherical, 
the shortest course would be a straight line, as used in plane 
sailing; but on the sphere the shortest course is a curve 
called a "great circle." Evidently, on all long voyages, the 
great circle course is the most advantageous one; that 
mariners do not more frequently use it is due to a peculiarity 
of their charts. 

1 This log is really too large by 10, so the subtraction is correct. 



38 NAVIGATION 

We cannot here enter into the details of chart "pro- 
jections," as the theory of chart making is called. It is 
sufficient to remark that a straight line drawn on the ordi- 
nary nautical charts (which follow the Mercator system), 
between any two ports, will not represent the shortest (or 
great circle) course between them. On such a chart, the 
great circle course between the two ports will appear to be 
longer than the straight line course, although it is really 
shorter. This accounts for the use of the longer Mercator 
course by many navigators. 

Now there is a kind of chart, called a "great circle sailing" 
chart, on which straight lines between ports really represent 
shortest (or great circle) courses. One would therefore 
naturally suppose that mariners would entirely discontinue 
the use of Mercator charts in favor of great circle charts. 
But there is a reason for not doing this. 

On Mercator charts, all terrestrial longitude meridians 
are represented by parallel vertical straight lines. Conse- 
quently, if we draw another straight line on 'the Mercator 
chart joining two ports, that line will make the same course 
angle (p. 10) with all the meridians. In this way, a navigator 
can get from a Mercator chart, by simply drawing a straight 
line, and quite without calculation, a course angle which will 
carry him from one port to another. And because the course 
angle so obtained is the same with respect to all meridians 
to be crossed by the ship it follows that the voyage can be 
completed (theoretically at least) from the one port to the 
other with the great advantage of never changing the course 
to be steered. 

On the other hand, the great circle track makes a different 
angle with every meridian it passes : so that the mariner 
must make very frequent changes in the course angle to be 
steered during the progress of a voyage. The simple 
Mercator track, without change of course, is called a "rhumb 
line" ; the serious objection to it is that it sometimes leads 
to greatly (and unnecessarily) lengthened voyages. 



DEAD RECKONING WITH LOGARITHMS 39 

The final conclusion is that Mercator charts, on account of 
their simplicity, are most convenient for short voyages, or 
for parts of long voyages when the land is not far away. 
But for shaping the main part of the course on a very long 
voyage, great circle sailing charts are to be preferred. 

At times, in order to avoid very high latitudes, or to round 
some projecting point of land, navigators must substitute for 
a single great circle track one "composed" of two or more 
shorter arcs of great circles. This is called "composite" 
sailing. 

Finally, for the sake of completeness, we shall merely 
mention two other kinds of sailing. " Parallel " sailing, which 
is simply middle latitude sailing when the latitude difference 
is zero; and "traverse" sailing, from which the traverse 
table gets its name. This is also the same thing as middle 
latitude sailing; but the special word "traverse" is used 
when the ship changes her course frequently, perhaps even 
during a single day. It is then possible to sum up the 
result of all the short courses which together make up the 
day's run. It is merely necessary to take from the traverse 
table the latitude difference and departure for each short course 
separately, and then to add 1 all the values of latitude differ- 
ence for a "summed latitude difference," and all the values 
of departure for a "summed departure." With these a 
"composite course and distance" can be taken from the 
traverse table, or calculated with logs, and these will repre- 
sent the motion of the ship, just as if she had steered an 
unchanged course during the entire day. 

1 It is necessary to sum separately latitude differences represent- 
ing northward motion of the ship and those representing southward 
motion. The difference of the two sums is what we need to know. 
The same is true of departures representing eastward and westward 
motion of the ship. 



CHAPTER IV 
THE COMPASS 

THE ship's course has been defined (p. 8) as the angle 
between the north and the direction in which the ship is 
sailing. To ascertain what this angle is, or, in other words, 
to steer the ship, mariners use the compass. The dial (or 
"card") of this instrument is divided, like any circle, into 
360. In the United States Navy these are numbered in 
such a way (fig. 7) that appears at the north, 90 at the 
east, 180 at the south, and 270 at the west. The numbers 
therefore increase in a "clockwise" direction. There are 
also compasses in which the numbering begins with at 
both the north and south points, and increases to 90 at the 
east and west points. But the United States Navy system 
of numbering is to be preferred. 

In addition to the above division and numbering, the dial 
is also divided into 32 points (pp. 10, 15), each containing 

ocn 

, or 11|. These points are then further subdivided 

o& 

into quarter points, all of which is shown clearly in Fig. 7. 

The naming of the points has not been done by chance, 
but in accordance with a definite rule. The four principal, 
or "cardinal," points are north, east, south, and west. The 
remaining points are located by a continued process of 
halving. Halfway between the cardinal points are the 
"inter-cardinal" points; and each is named by combining 
the names of the two cardinal points adjacent to it. Thus 
northeast (abbreviated N.E.) is halfway between north 
and east. Again halving and combining names, we get 
points like E.N.E., S.S.E., etc. Still once more halving 
completes the tally of 32 points : but a combination of 
names would now be too complicated. However, since 

40 



THE COMPASS 



41 



each of these final points must necessarily be adjacent to a 
cardinal or inter-cardinal point, they are named by simply 
increasing the name of such adjacent cardinal or inter- 
cardinal point. This is accomplished with the word "by." 




FIG. 7. Compass Card. 

Thus we find, adjacent to N.E., the points N.E. byE., and 
N.E. by N. In the light of the above, it is easy to "box" 
the compass, as seamen say, or to name the 32 points in 
order. 

When the point system of division is used, and an accuracy 



42 NAVIGATION 

closer than a single point is required, the compass card is 
still further subdivided into quarter points. In naming 
these it is customary, in the United States Navy, to "box" 
from N. and S. towards E. and W. Thus the space between 
N.N.E. and N.E. byN. would be divided into four parts 
thus : N.N.E.iE., N.N.E ^E., N.N.E.f E. But an excep- 
tion is made to this last rule' in the case of quarter points 
adjacent to a cardinal or inter-cardinal point. These last 
are always put first in naming the quarter points. Thus, 
between E. by N. and E., if we always boxed from N. towards 
E., we should have : E. by N.|E., E. by N.^E., E. by N.f E. 
But it is customary, because shorter, to name these quarter 
points E.fN., E.N., and E.|N. 

Inside the "bowl" of the compass, and adjacent to the 
card, a black line is marked on the bowl. This line is in 
plain view of the steersman, through the glass cover of the 
compass, and is called the "lubber line." When the ship 
is headed in such a way that this line comes opposite N.E., 
for instance, on the card, the ship will be on a N.E. course, 
which makes an angle of 45 with the north. 

But would the ship really be traveling on a line making 
a 45 angle with the geographic meridian, or direction of 
the north pole of the earth? She would be doing so only 
if the compass were absolutely correct. This is practically 
the case with the "gyro-compass," a mechanical contrivance 
now much used in the navy, but not the case with the ordi- 
nary "magnetic" compass. 

In Chapters II and III, concerning dead reckoning, we have 
always used the word "course" as if all compasses were 
absolutely correct. But since they are not correct, it is 
now necessary to make allowance for their errors. In other 
words, whenever we use a compass, we must first ascertain 
the difference between the "true course" and the "compass 
course." It must not be supposed from this statement that 
a ship can be steered on two different courses at the same 
moment. There is really only one direction along which 



THE COMPASS 43 

the ship is moving : but the angle between that direction 
and the true north may be different from the angle between 
it and the "compass north." It is the course measured 
from the true north that must be used in all dead-reckoning 
calculations, and that always results from such calculations : 
but for steering the ship by means of a compass the steers- 
man must be furnished with the course as measured from 
the compass north. Therefore it is essential for the navigator 
to know the difference between the two. This difference 
is called the "error" of the compass. 

Unfortunately, this error is made up of two parts. The 
first, called "variation" of the compass, is due to peculiari- 
ties in the earth's magnetism, and is quite different in dif- 
ferent places on the earth. It also varies in different years 
at the same place. But at any one time, all ships in the 
same part of the ocean will have the same variation. 

The mariner can always ascertain how great the varia- 
tion is in his part of the ocean, because it is always marked 
on his chart. Certain curved lines are drawn on the chart ; 
and if the ship is located on or near a line marked "varia- 
tion 10," for instance, it follows that the navigator must 
on that day allow for 10 of variation. It is also important 
to take into consideration possible changes in the variation. 
Sometimes the annual change is marked on the chart; if 
not, it is important to use a chart of recent date. 

The second part of the error is called "deviation" and is 
due to peculiarities in the magnetism always developed in 
the metallic parts of the ship itself. It is different in dif- 
ferent ships, even in the same part of the ocean, and is even 
different in the same ship, when she is headed on different 
courses. Methods have been invented for "compensating" 
marine compasses, so as to remove the effects of deviation, 
and these methods are quite effective. But even when 
they are used, it is necessary, before beginning a long voyage, 
to have a "compass adjuster" visit the ship. He will then 
"swing" the ship on a number of different courses, and 



44 NAVIGATION 

adjust the compass so that it will be as nearly correct as pos- 
sible. Finally, he will determine, by means of astronomic or 
other observations, just what the remaining compass devia- 
tion is on all the various courses, and give the navigator a 
table of these remaining deviations. This table must be taken 
into account in "shaping" the ship's course during the 
voyage. The navigator must also, from time to time, check 
these tabular deviations while at sea by means of astronomic 
observations of his own, to take care of possible changes. 

Such astronomic observations are made with an instru- 
ment (the "azimuth circle"), which can be attached to the 
compass, and with which the "compass bearing" of the 
sun or any other object can be observed. The compass 
bearing is simply the compass direction of the object, as 
seen from the ship ; or the compass course on which the ship 
would be steered, if she were moving directly toward the 
object. When the sun is used, its true bearing, measured 
from the true north, can be taken from astronomic tables 
which will be explained later; and it is called the sun's 
"azimuth." A comparison of this true bearing with that 
measured on the compass with the azimuth circle then makes 
the compass error known. 

When it is not convenient to observe the sun, it is possible to 
substitute observations of a distant well-defined terrestrial ob- 
ject, whose true bearing can be measured on a chart for com- 
parison with various compass bearings observed while the ship 
is being swung. Another method is to set up a compass on 
shore, away from any iron or steel, and use it to determine 
the bearing of the distant object. And there is still another 
method, if the above compass and the ship's compass are inter- 
visible. For the bearing of each may then be taken from the 
other, and these should differ by exactly 180. If they do not, 
the variation from 180 must be due to deviation on board. 

The "pelorus" is another instrument which may at times 
replace the azimuth circle. It is located anywhere on the 
ship, at a convenient point for observation, and not neces- 



THE COMPASS 



45 



sarily close to the compass. It has a "dummy card" and a 
lubber line. The dummy card can be turned until the 
lubber line indicates the same course as the real compass. 
Observations of bearings with the pelorus will then obviously 
be the same as if made on the compass with the azimuth circle. 
The advantage of the pelorus is that it can be used anywhere 
on board, while the compass must be kept constantly in the 
exact place where it was "adjusted" before leaving port. 

The error thus determined astronomically or otherwise 
is the sum of the variation and deviation. If we indicate 
by E the total compass error in that place, at that time, on 
that ship, and on that course ; by D the deviation similarly 
described ; by V the variation at that time and in that place ; 
and if all three are counted from in the usual direction 
around the compass card, then 
we have the formula : 

(1) E = V + D. 

By counting in the usual direc- 
tion, we mean counting from the 
north around to the east, as all 
courses are counted (p. 19) ; so 
that a compass error of 10. for 
instance, would mean that the 
compass north pointed 10 east 
of the true north, or had a true 
bearing of N. 10 E. (p. 19). 
This is shown in Fig. 8, which 
also shows the ship's course, 
counted in the same way. 

It is clear from the figure that if we now indicate : 

by C, the ship's compass course, 

by T, the ship's true course, 

by E, the compass error, 

we shall have the formula : 




FIG. 8. Compass Error. 



(2) 



= C + E. 



46 NAVIGATION 

The simple formulas (1) and (2) enable the navigator to 
make all necessary compass calculations. The following 
are examples. 

Suppose, for instance, that the error E has been deter- 
mined by observation, and the variation V taken from the 
chart. Formula (1) then makes it possible to calculate 
the deviation D. For the formula shows that E is the sum 
of V and D ; and so D must be the difference of E and V, 

or: D = E - V. 

Thus the deviation D becomes known, as a check on the 
compass adjuster's work, and, while this value of D is cor- 
rect only for the particular course on which the ship was 
headed at the time the observation was made, yet that 
course is the very one for which it is especially important 
to have correct information. 

Again, suppose dead-reckoning calculations show that the 
ship is to sail on a 40 course. These calculations always 
furnish the true course (p. 43) so that T = 40. The 
variation being known from the chart, and the deviation 
from the adjuster's table, we know from (1) E = V + D. 
Then from (2) we see that C = T E, which gives the 
compass course. Let us suppose in the present case, that 
V was 9, D 1 ; then E = V + D = 9 + 1 = 10 ; and 
since T = 40, C = T - E = 40 - 10 = 30 ; and the 
helmsman would be directed to steer a 30 course by com- 
pass. 

If, in Fig. 8, the compass north happened to be 10 on the 
left side of the true north, instead of the right, the error E 
would be 350, instead of 10 (see also fig. 7, p. 41). This 
might be made up of a variation V of 349 and a deviation 
D of 1, as before. If the true course is again to be 40, 
the compass course would be 40 350, according to the 
formula C = T E. This subtraction being impossible, 
we increase the 40 by a complete circumference of 360j 
which is always permissible, and then have : 



THE COMPASS 47 

C = 360 + 40 - 350 = 50. 

The ship would be steered on a compass course of 50. 

An alternative way to take care of errors, variations, 
and deviations on the left side of the true north is to mark 
them with the negative or minus sign. Instead of calling 
V 349, we might call it 11. This is really the best way, 
and leads to the same result as before, if we remember that 
the subtraction of a minus quantity is always equivalent to 
an addition. In the example just given, calling V 11, 
instead of 349, we should have : E = F + D = - 11 + 
1 = - 10; and C = T - E = 40 - (- 10) = 50, the 
same compass course as before. 

An older way of designating variations, deviations, and errors 
is to call them east when the compass north points to the 
right of the true north, and west when it points to the left 
of the true north. This method leads to the necessity of 
providing various rules or diagrams with which to make 
compass calculations. We think the best way to avoid 
error (and such errors may lose ships and lives) is to use the 
method here given with its two simple formulas. When 
some other designation of the error, or some other method 
of numbering the card, is demanded by a captain, it is always 
possible to conform to that demand, but also to translate 
every problem into our method "(in imagination at least) 
as a check against mistake. 

The following is an example of a compass adjuster's "devia- 
tion table," taken from Bowditch's " Navigator " (1916 
edition). The deviations are set down in degrees and tenths 
of a degree, instead of degrees and minutes, for convenience 
in the further calculations. The ship was swung so that 
her head bore successively around the horizon, and obser- 
vations were made at intervals of 15. This is a smaller 
interval than is usually necessary ; and the deviations in the 
table are much larger than commonly occur in a modern 
well-compensated compass. 



48 



NAVIGATION 



DEVIATION TABLE 



BEARING 




BEARING 




BEARING 




BEARING 




OF SHIP'S 


DEVIA- 


OP SHIP'S 


DEVIA- 


OF SHIP'S 


DEVIA- 


OF SHIP'S 


DEVIA- 


HEAD BY 


TION 


HEAD BY 


TION 


HEAD BY 


TION 


HEAD BY 


TION 


COMPASS 




COMPASS 




COMPASS 




COMPASS 




o 


o 





o 


o 


o 


o 


o 





- 15.5 


90 


- 9.1 


180 


+ 17.9 


270 


+ 9.9 


15 


- 14.9 


105 


-9.0 


195 


+ 23.8 


285 


+ 1.9 


30 


- 13.3 


120 


- 7.8 


210 


+ 27.1 


300 


- 4.2 


45 


- 11.3 


135 


- 5.9 


225 


+ 25.6 


315 


- 10.3 


60 


- 10.0 


150 


-2.3 


240 


+ 22.0 


330 


- 13.6 


75 


- 9.7 


165 


+ 8.5 


255 


+ 15.9 


345 


- 16.0 



To illustrate the use of this table, let us suppose the ship 
to be sailing on a compass course of 165, in a part of the 
ocean where the variation is + 10, or 10 E. Using formula 
(1) (p. 45), and finding from our table that the deviation D 
for 165 is + 8.5, we have the compass error E = V + D 
= + 10 + 8.5 = + 18.5. By formula (2) (p. 45) the true 
course of the ship is T = C + E = 165 + 18.5 = 183.5. 
We should use this true course 183.5 in calculating later 
the ship's position by dead reckoning (p. 10). 

If the compass variation were everywhere the same, it 
would be more convenient to have a table of compass errors, 
instead of a table of deviations ; but because the variation, 
as given on the chart, varies greatly, the table must be 
specially made for deviations only. 

Equally important with the above use of our deviation 
table is its inverse use. When the navigator has calculated 
by dead reckoning the course he must steer, that course, 
as it comes from the calculations, will be a true course (p. 
43) ; and it is necessary to turn it into a compass course for 
the use of the steersman. 

To do this we must know the deviation ; and we cannot 
get it directly from the deviation table above, because the 
use of that table presupposes a knowledge of the compass 
course, the very thing we are trying to find. The best 



THE COMPASS 



49 



way to avoid this difficulty is to imagine the deviation to be 
non-existent, for the moment, and to make use of the "mag- 
netic course," defined as the course which would be indi- 
cated by the compass, if deviation were thus totally absent. 
Under these circumstances, formula (1) gives E = V, since 
D = ; and if we designate the magnetic course by M , we 
may write, in place of formula (2) (p. 45) : 

(3) M=T-V. 

Let us suppose a case in which the variation is + 10, and 
the desired true course of the ship 175. Then the magnetic 
course, allowing for variation only, will be, by formula (3) : 

M = T - V = 175 - 10 = 165. 

This course is not really a compass course, because no 
account has yet been taken of the deviation. Nor can we 
yet find the deviation directly from the deviation table, 
because in that table we must still know the compass course 
to use as the argument (p. 10), whereas we know as yet only 
the magnetic course. Therefore navigators should always 
request the compass adjuster to furnish a "second deviation 
table," in which the argument is the magnetic course, in- 
stead of the compass course. Such a second table can al- 
ways be calculated from the other. We here give one that 
has been calculated from the table on the preceding page. 

SECOND DEVIATION TABLE 



MAG- 




MAG- 




MAG- 




MAG- 




NETIC 




NETIC 




NETIC 




NETIC 




BEARING 


DEVIA- 


BEARING 


DEVIA- 


BEARING 


DEVIA- 


BEARING 


DEVIA- 


or SHIP'S 


TION 


OF SHIP'S 


TION 


OF SHIP'S 


TION 


OF SHIP'S 


TION 


HEAD 




HEAD 




HEAD 




HEAD 













o 


o 


o 





o 





- 14.9 


90 


-9.0 


180 


+ 11.0 


270 


+ 16.5 


15 


- 13.4 


105 


- 8.4 


195 


+ 16.9 


'285 


+ 4.1 


30 


- 11.7 


120 


- 6.9 


210 


+ 21.3 


300 


- 7.1 


45 


- 10.4 


135 


' -4.8 


225 


+ 24.9 


315 


- 13.2 


60 - 


- 9.8 


150 


- 1.4 


240 


+ 26.8 


330 


- 15.7 


75 


- 9.3 


165 


+ 5.0 


255 


+ 24.1 


345 


- 15.5 



50 NAVIGATION 

We also add as an example the calculation of one number 
in the second table from those given in the first. We shall 
find the deviation corresponding to the magnetic course 
165 ; and we do it by a kind of interpolation (p. 12). From 
the first table we have the deviation 2.3 for the compass 
course 150. Since the deviation is the only difference 
between compass and magnetic courses, it follows that 
150 2.3, or 147.7 magnetic, corresponds to 150 by com- 
pass. Similarly, 173.5 magnetic corresponds to 165 by 
compass. 

The magnetic course 165 for which we are making the 
calculation falls between 147.7 and 173.5, and exceeds 
the smaller of the two by 17.3. The whole difference be- 
tween 147.7 and 173.5 is 25.8. Similarly, the whole dif- 
ference between the two compass courses involved is 15. 
Therefore we may write the proportion : 

25.8 : 15 = 17.3 : x, 

where x is the excess over 150 of the compass course corre- 
sponding to 165 magnetic. 

Solving this proportion by the ordinary rules of arithmetic, 
we have : 

= 15 X 17.3 = 1QO 
25.8 

The compass course belonging to 165 magnetic is there- 
fore 150 + 10.0 = 160.0. The corresponding deviation 
is 165 - 160.0 = + 5.0, 1 which is therefore the deviation 
for 165 magnetic, and appears as such in the second table. 
This entire table can be computed from the first table in an 
hour. 

Sometimes the second deviation table gives compass courses 
instead of deviations. It is then often called a " table of 

1 A comparison of formulas (1), (2), and (3) shows that 
D = M C ; so that the deviation is obtained by subtracting the 
compass course from the magnetic course. This is also evident 
from the definition of a magnetic course (p. 49). 



THE COMPASS 51 

steering courses " ; and in the example just calculated it 
would give the compass or steering course 160 for the mag- 
netic course 165, instead of giving the deviation + 5. 

We shall still further illustrate this important matter by 
an example, supposed to occur on board a ship for which 
our two deviation tables hold good. 

What is the compass course to be given the helmsman at 
Sandy Hook, on a voyage to St. Vincent? 

We have already found, from dead-reckoning calculations 
(p. 22) the course 119. Being the result of a dead-reckon- 
ing calculation, this is a true course. The track chart of 
the north Atlantic gives the variation at Sandy Hook as 
10 W., or - 10. The true course being 119, we get the 
magnetic course, allowing for variation only, by formula (3), 
M = T - V = 119 -(- 10) = 129. The second devia- 
tion table shows that : 

for magnetic course 120, the deviation is 6.9, and 
for magnetic course 135, the deviation is 4.8. 

Magnetic course 129 falls between 120 and 135, so that 
an interpolation (to be extremely exact) between 6.9 
and 4. 8 makes the deviation for magnetic course 129 
come out 5.6. Formulas (1) and (2) now give : 

Error =E = V+D = -W- 5.6 = - 15.6 

Compass course = C = T-E = 119 -(- 15.6) =134.6. 

To check this, we can now solve the same problem in the 
inverse way with the first deviation table. For the compass 
course 134.6, this table gives the deviation as 5.9. The 
variation being 10, we have : 

E = V + D = -10 - 5.9 = - 15.9 and 
T = C + E = 134.6 - 15.9 = 118.7, 

agreeing very closely with the true course 119, with which 
we started. This shows that the two deviation tables are 
quite consistent in this case, and also checks the accuracy 
of the calculation. 



52 



NAVIGATION 



We shall close this chapter with the following little table, 
showing the correspondence between the two methods of 
dividing the compass card into points, and into degrees. 

COMPASS POINTS AND DEGREES 





- , 




o , 




o , 




o , 


North 





East 


90 


South 


180 


West 


270 


N. by E. 


11 15 


E. by S. 


101 15 


S. by W. 


191 15 


W. by N. 


281 15 


N.N.E. 


22 30 


E.S.E. 


112 30 


s.s.w. 


202 30 


W.N.W. 


29230 


N.E. by N. 


33 45 


S.E. by E. 


12345 


S.W. by S. 


21345 


N.W. byW. 


303 45 


N.E. 


45 


S.E. 


135 


S.W. 


225 


N.W. 


315 


N.E. by E. 


56 15 


S.E. by S. 


146 15 


S.WibyW. 


236 15 


N.W. by N. 


326 15 


E.N.E. 


6730 


S.S.E. 


157 30 


W.S.W. 


24730 


N.N.W. 


33730 


E. by N. 


7845 


S. by E. 


16845 


W. by S. 


25845 


N. by W. 


34845 



J pt. = 2 49' 



i pt. = 5 38' 



pt. = 8 26' 



1 pt. = 11 15' 



CHAPTER V 
COASTWISE NAVIGATION 

BEFORE proceeding to a consideration of navigation by 
means of astronomic observations, as it is practiced on the 
high seas, we must first explain certain methods by which 
it is possible to ascertain a ship's position in latitude and 
longitude while she is in sight of land. Often such methods 
suffice to complete a long coastwise voyage in safety; they 
are always important for a last determination of the ship's 
position before a deep-sea voyage actually begins. Such a 
last determination is called "taking a departure" (cf. p. 2), 
and from such point of departure dead-reckoning calcula- 
tions begin for the first day of the voyage. 

Any determination or fixing of a ship's position, by astro- 
nomic observations or otherwise, is often called, for brevity, 
a "fix." To obtain one while in sight of land it is customary 
to make observations upon well-known objects ashore, 
such, for instance, as lighthouses, or other conspicuous 
objects marked on the chart. It is always possible to ob- 
serve the bearings of such objects from the ship's deck with 
the compass, azimuth circle, or pelorus (p. 44). 

When there is but one such object in sight, it is impossible 
to secure a fix with ordinary instruments, if the vessel is 
at anchor. But if she is running, it is merely necessary to 
take two bearings, and to estimate. the distance run by the 
ship in the interval between the two. Figure 9 will make 
this matter clear. A lighthouse ashore is at L. SS" is the 
direction of the ship's course; S her position when the 
first bearing was observed, and S' her position at the time 
of the second bearing. SN is .the direction of the north. 

53 



54 NAVIGATION 

After taking the first bearing, the navigator must calculate 
ythe angle S"SL, between the ship's course SS" and the 

lighthouse direction SL. Thus, 
if the ship's course angle NSS" 
(p. 10) was 20, and the bearing 
NSL was 42, the angle S"SL 
would be 42 - 20 =22. As 
the ship proceeds on her course, 
the angle S"SL will become 
larger, and a second bearing must 
be taken at the moment when 
the ship reaches the point S', 
where the angle S"SL has become 
S"S'L. This point S' must be 
so chosen that the angle S"S'L 
is just twice the angle S"SL ob- 
served at S ; or, in this case, 44. 
This is called " doubling the bear- 
FIG. 9. Ship's Position by Two ing from the bow/' and it can 

easily be accomplished if we con- 
tinue watching the compass bearing of L as the ship goes 
ahead, and catch the observation at the right moment. The 
ship's course not having been changed from 20 (this is 
important), the right moment will occur when L bears 
20 + 44 =64 by the compass. 

It can easily be proved by geometry that the distance 
*S'L between the ship at S' and the lighthouse at L will be 
equal to the distance SS' traveled by the ship in the inter- 
val between the two observations. This distance can be 
estimated quite accurately with an instrument called a 
"log," or "patent log," which is towed astern of the ship. 
It is so constructed that it turns as it is pulled through the 
water, and the number of turns is automatically counted by 
an attached contrivance on deck. The count is (also auto- 
matically) turned into miles of distance ; so that the log on 
deck will indicate how far the ship traveled from S to S'. 




COASTWISE NAVIGATION 55 

As soon as we know the distance S'L and the bearing of 
the line S'L, we can "lay down" or "plot" the position of 
S' on the chart; and this will be a "good fix." To do this, 
let us indicate by B' the bearing of the line .S'L, and then 
draw on the chart, through the lighthouse L, a pencil line 
whose bearing from L is B' + 180, or "B f reversed." This 
can be done with a "course protractor," or with "parallel 
rulers," instruments to be purchased from any dealer in 
navigators' supplies. Next we measure or "lay off" on that 
line the distance S'L, equal to the run SS' as it came from 
the log. We always know the right "scale" of the chart 
(or fraction of an inch corresponding to one logged mile) 
which must be used in laying off the distance S'L; for we 
know that one mile always corresponds to 1 minute of 
latitude (p. 15), and the right- and left-hand edges of the 
chart are always divided into degrees and minutes of latitude. 

Since the above bearings were observed by compass, it 
is now important to consider the compass error (p. 43). 
This will not affect the observations, because it will be the 
same for both ship's course and lighthouse bearing, so the 
angles S"SL and S"S'L, which are obtained by subtraction, 
will be the same as if there were no compass error. But 
when we come to plotting on the chart, the compass bearing 
B' must be corrected by adding the deviation from the 
deviation table (pp. 48, 49). The resulting magnetic bear- 
ing (p. 49) must be used for B', if the chart has printed 
on it a compass card (p. 41) showing magnetic bearings. 
If the printed card shows true bearings only, B' must be 
corrected for both deviation and variation (p. 43). 

A specially important case of the foregoing occurs when 
the two angles S"SL and S"S'L are 45 and 90. The 
second bearing B' will then put the light just abeam, and 
the distance by log, SS', is the distance, at which the ship 
passes the light abeam. This case is called a "bow-and- 
beam bearing." The navigator sights the light when it bears 
45 or 4 points (p. 52) "broad" on the bow, "starboard," 



56 NAVIGATION 

or "port." He then "reads" the log. When he brings 
the light abeam through the motion of the ship, he reads 
the log again, and the run in the interval, as taken from the 
log, is the light's distance abeam. 

When sailing along the coast, it is particularly important 
so to shape the ship's course that lights and other promi- 
nent landmarks will be passed at the right distance abeam. 
The chart shows what the right distance is : if the navigator 
shapes a course which makes the distance abeam too small, 
he may fail to clear rocks or shoals extending seaward ; and 
if he makes it too large, he may lengthen his voyage unneces- 
sarily in rounding the light. 

There are certain pairs of angles (S"SL and S"S'L) which 
will make known the coming distance abeam long before 
the ship is dangerously near the light. These angles, S"SL 
and S"S'L, are called "bearings from the bow" (see p. 54), 
since they are really measured from the ship's bow instead 
of the north. If the two bearings from the bow are either 
of the following pairs : 

22 and 34, 32 and 59, 

27 and 46, 40 and 79, 

then the logged distance in the interval between the two 
observations is the distance at which the ship will pass the 
light abeam if she continues on her present course. This 
kind of observation will inform the navigator whether his 
course is safe in ample time to change it if necessary ; and, 
since in this case no bearings are marked on the chart, no 
attention need be paid to compass error. 

When two or more known and conspicuous landmarks 
are visible from the ship, it is possible to secure a fix by 
means of "cross-bearings." Observe the bearings of the 
objects as nearly simultaneously as possible. Allow for 
compass error in the manner just explained. Calculate 
for each object a reversed bearing by adding 180 to its 
observed bearing. Draw on the chart through each object 



COASTWISE NAVIGATION 



57 



a pencil line having the proper reversed bearing and these 
lines will intersect at the point on the chart where the ship 
is located. Figure 10 
illustrates this matter. 
L, L', L" are lights or 
landmarks ashore, 
visible from the ship, 
and also printed on 
the chart. The ship 
is at S. The lines in- 
tersecting at S repre- 
sent the reversed 
bearings of L, L', L", 
as observed from S. 
Only two lines are nec- 
essary ; and they 
should be chosen so 
that the angle be- 
tween them is as near 




FIG. 10. Ship's Position by Cross Bearings. 



a right angle as possible, if high accuracy is required in the 
fix. The third object and line merely serve as an additional 
check or safeguard against error. 

In addition to the foregoing methods of locating a ship 
by observations of objects ashore, there is a way to avoid 
sunken rocks or shoals without actually locating the ship 
on the chart. It is called the "danger angle," and is shown 
in Fig. 11. The small circle is supposed drawn on the chart 
around a rocky shoal K which must be cleared by the ship 
traveling along the course SS f . To make certain of clearing 
it safely, the navigator selects two visible objects ashore, 
and shown on the chart at L and L'. He draws on the 
chart a large circle passing through L and L', and just touch- 
ing the dangerous small circle at T. There is no difficulty 
in finding the center of the large circle, because it must be 
somewhere on the line PQ, which is drawn at right angles 
to the line LL' at its middle point P. A few trials with a 



58 



NAVIGATION 



pair of compasses will locate the center. Next, the two lines 
LT and L'T are drawn. Then the angle LTL' is called the 
danger angle. 

Now it is a principle of geometry that if we select other 
points on the large circle, such as T' and T", the angles 




FIG. 11. The Danger Angle. 

LT'U, LT"L', etc., will all be equal, and will contain the 
same number of degrees as the danger angle LTL'. It fol- 
lows that if the navigator measures from the deck the angle 
formed by two lines drawn to the ship from L and L', and 
if he finds it equal to the danger angle LTL', as measured 
on the chart with a protractor (p. 55), he then knows that 
the ship is somewhere on the large circle, and is therefore 
perhaps too near the small dangerous circle. If, on the 
other hand, the ship is entirely outside the large circle, and 
therefore surely safe from the gangers of the small circle, 



COASTWISE NAVIGATION 



59 



the measured angle at the ship between the objects L and 
L' will always be smaller than the danger angle LTL'. 

Angles can be measured from the deck by taking compass 
bearings of L and L'. The difference of the two will be the 
deck angle, which should be smaller than the danger angle 
measured on the chart. But the very best way to measure 
the deck angle is to use the sextant, an angle-measuring 
instrument to be described later (p. 61). 

The danger angle can also be used when it is necessary to 
pass between a sunken danger circle and the shore. The 
large circle is then drawn through L and L' as before, but in 
such a way as just to touch the inside of the small circle 
instead of the outside. To pass inshore of the small circle 
it is then necessary for the navigator 
to keep his measured deck angle larger 
than the danger angle, instead of 
smaller. 

Navigators also use at times a 
means of safety known as the " danger 
bearing," illustrated in Fig. 12. 
There is but one charted object in 
sight ashore at the point L. The ship 
at S must steer in such a way as to 
avoid sunken rocks at K. Evidently, 
she must pass outside the line SQ, of 
which the bearing from the north is 
the angle NSQ, which can be meas- 
ured on the chart. This is the danger 
bearing, and the ship's course SS', to 

be safe, must be greater than the danger bearing. In the 
case shown in the figure, the danger bearing would be very 
useful long before a fix could be had by means of bearings 
from the bow or bow-and-beam bearings. 

Finally, to complete this part of our subject, it is neces- 
sary to mention "soundings," which are a method of feel- 
ing the land, even when it cannot be seen. By means of 




FIG. 12. The Danger 
Bearing. 



60 NAVIGATION 

the "lead-line" the mariner can ascertain when he is in 
shoal water ; and as depths of water are always marked on 
the chart, he can often get valuable information as to the 
ship's position. As she runs along her course, he can take 
a "line of soundings" and upon examining the chart he 
will often find but a single possible line on the chart where 
the charted depths correspond with those observed. It 
follows that the ship's course must have been along that 
line on the chart ; and at an anxious moment, in a fog, such 
a check will be a great relief to the navigator. Even in 
the ocean, far from land, it is possible to take soundings 
with the "sounding machine" at great depths, and in some 
parts of the ocean quite accurate locating of the ship will 
result. Specimens from the ocean floor can also be brought 
up by attaching some sticky grease to the bottom of the 
lead, and at times these specimens also give information 
of value, for the charts always specify the kind of bottom 
existing in various parts of the ocean. 



CHAPTER VI 
THE SEXTANT 

WE have twice made reference to this instrument once 
(p. 5) as a contrivance for ascertaining by observation how 
high the sun is in the sky, and again (p. 59) in the measure- 
ment of the danger angle. These two uses of the sextant 
are not inconsistent, for it is really intended for the measure- 
ment of any angle (p. 8) formed at the observer's eye by 
two lines drawn to two distant objects. In the case of the 
danger angle these two distant objects are landmarks 
ashore; in the case of the sun they are the "horizon" line 
(where sea and sky seem to meet), and the sun itself. This 
height of the sun (or of any star) in the sky is called its 
" altitude"; and so the altitude is always an angle, to be 
measured in degrees and minutes. The point directly over- 
head is the "zenith"; the angle between lines drawn to 
horizon and zenith is 90, or a right angle. An altitude of 
40, for instance, simply means that the distance from the 
horizon to the sun is f$ of the total distance from horizon 
to zenith. 

Figure 13 will give an idea of the construction of the sex- 
tant. 1 The essential parts are two small silvered mirrors, 
M and ra; a telescope, EK; and a circle, A A, engraved 
with "graduations," by means of which angles may be 
measured upon it in degrees, minutes, and seconds. The 
mirror m and the telescope EK are firmly attached to the 
sextant ; but the mirror M is pivoted in such a way that it 

1 Quoted in part from Jacoby's " Astronomy, a Popular Hand- 
book," Macmillan, 1913 ; reprinted 1915. 

61 



62 



NAVIGATION 



can be turned, and the angle through which it is turned 
measured on the circle by means of the index CB. When 
the mirror M is turned until it is parallel to the fixed mirror 
m, the circle "reads" or indicates 0, because the angle be- 
tween the two mirrors is then 0. In all other positions 




FIG. 13. The Sextant. 

of the mirror M the circle measures the angle between the 
two mirrors. P and Q are sets of colored glasses, which can 
be interposed temporarily, when the sun's rays are so bril- 
liant as to be hurtful to the observer's eye. R is a small 
magnifying glass, pivoted at S, intended to facilitate the 
examination of the index CB. At C and B are shown the 
"clamp," by which the index can be fastened to the circle, 
and the "tangent screw," or "slow-motion screw" which 
will adjust it delicately, after it has been clamped. / and F 
are additional telescopes or accessories. 

The mirror m has an important peculiarity. The silver- 
ing is scraped away at the back of the mirror from half its 



THE SEXTANT 63 

surface. Thus only one half reflects ; the other half is 
simply transparent glass. A navigator looking into the 
telescope at E will therefore look through the mirror m with 
half his telescope, and with the other half he will look into 
the mirror. 

Now it is a fact that half a telescope acts just like a whole 
one. If a person using an ordinary spy-glass half covers 
the big end with his hand, he will see the same view he saw 
with the whole glass. Only, as half the "light-gathering" 
power is cut off, this view will be fainter, less luminous. 
Applying this to the sextant telescope, it is clear that the 
observer will see two things at once : with half the telescope 
he will see what is visible through the mirror m; and with 
the other half he will see what is visible by reflection from 
the mirror m. 

If he holds the sextant in such a position that the telescope 
is horizontal, while the frame of the instrument is vertical, 
he will see the visible sea horizon with half the telescope 
through the mirror m. If the other mirror M is then turned 
to the proper position, it is possible to see the sun in the sky 
at the same time, with the other half of the telescope, the 
solar rays having been reflected successively from both mir- 
rors, M and m. To make this possible, the sextant tele- 
scope must be aimed at that point of the sea horizon which 
is directly under the sun. The solar rays will then strike 
the mirror M first ; be thence reflected to the silvered part 
of the mirror m; and finally reflected a second time into 
the telescope. Therefore the observation consists in so 
turning the movable mirror M, that the sun and horizon 
can be seen coincidently in the telescope. 

The angle between the mirrors can then be measured 
on the circle ; and it is easy to prove by geometry that the 
angular altitude of the sun will be twice the angle between 
the two mirrors. Thus it should merely be necessary to 
double the mirror angle, as indicated by the sextant index, 
to obtain the solar altitude. But the sextant makers always 



64 NAVIGATION 

save the navigator the trouble of doubling the angle by the 
simple device of numbering half degrees on the arc AA as 
if they were whole degrees ; so the angle as it comes from the 
sextant is already doubled for further use. The mirror m 
is called the "horizon glass," because the navigator looks 
through it at the horizon. The other mirror M is the "index 
glass," because it is attached to the index arm. 

When the sextant is used for non-astronomical observa- 
tions, such as the danger angle, the frame is held horizontally, 
instead of vertically, as in observations of the sun. The 
telescope is aimed at the left-hand object ashore, and that 
object is viewed through the horizon glass m. The index 
glass M is then turned until light from the right-hand object 
is also brought into the telescope, after successive reflections 
from the two mirrors M and m. The two objects will then 
be seen "superposed," and the sextant arc will give the 
angle between two lines drawn from the observer on board 
to the two objects ashore. This angle should be smaller 
than the danger angle to keep the ship safely off-shore of 
sunken dangers (p. 59). 

Reading the sextant circle, or ascertaining from it the 
angle that has been measured, is accomplished by means of 
a "vernier." This is a short circular arc, engraved with 
graduations resembling those on the sextant circle, attached 
to the index CB (fig. 13) just under the little magnifier R. 
It is so placed that the graduations on the sextant circle 
and the vernier are close together and can be seen at the 
same time through the magnifier R. Figure 14 gives an idea 
of the vernier and a part of the sextant circle near the zero 
of its graduations. Numbers on both circle and vernier 
increase toward the left. On the circle, the largest spaces, 
marked by long lines, are whole degree spaces. Each is 
usually divided into two halves of 30' each indicated by 
shorter lines, and these are again subdivided into three 
small spaces of 10' each. The divisions on the vernier 
resemble those on the circle, except that the degree spaces 



THE SEXTANT 



65 



of the former are here called min- 
ute spaces, and the 10' spaces of 
the former are called 10" spaces. 
The real index of the instru- 
ment is the zero mark on the 
vernier, sometimes provided with 
an engraved "arrow." If this 
falls exactly on a degree mark of 
the circle, say the 1 mark, the 
reading of the instrument is ex- 
actly 1 0' 0". If it falls exactly 
on a small line of the circle, say 
the second to the left of the 1 
mark, the reading is exactly 1 
20' 0". But if it falls between two 
of the small lines, say between 
the 20' and 30' marks to the left 
of the 1 mark (as shown in the 
figure), the reading must be 1 
20' and a "bit." It is the busi- 
ness of the vernier to estimate 
the size of that bit. To do this 
look along the vernier until you 
find a line which is exactly op- 
posite some line on the circle. 
There will always be such a line : 
in the figure it is the 6' line of the 
vernier. Pay no further atten- 
tion to noting which line on the 
circle is the one thus " exactly 
opposite"; it matters not which 
line it is. But read carefully the 
number on the vernier belonging 
to the "exactly opposite" line 
you have found there. Being on 
this occasion the 6' line, it follows 





66 NAVIGATION 

that the bit is 6' ; and as we found the reading to be 1 20' 
and a bit, the complete reading is 1 20' + 6' = 1 26'. 

If the vernier line that happened to be "exactly opposite" 
was not one of the ten long minute lines, but fell between 
two of them, it would indicate that the bit was made up of 
minutes and seconds, instead of being an exact number of 
minutes. For each space the "exactly opposite" vernier 
line happens to lie to the left of a long vernier minute line, 
10" must be added to the bit. For instance, if in the figure 
the "exactly opposite" vernier line was the next short one 
to the left of the 6' long line, the bit would be 6' 10", and the 
complete reading 1 26' 10", instead of 1 26'. But seconds 
are not really required when observing aboard ship, so that 
it will be sufficient, in using the vernier, to find the number 
of the long vernier line that comes nearest to being "exactly 
opposite." 

It will also be noticed in the figure that the sextant circle 
has some additional graduations to the right of the mark. 
These are called "off the arc" graduations, and it is some- 
times necessary to read a small angle upon them, measuring 
from the mark to the right instead of the left. This makes 
it necessary to read the vernier backwards, calling the 0' 
mark of the vernier 10' and the 10' mark 0'. 

This backward reading of the vernier offers no particular 
difficulty, and it is especially useful in determining by ob- 
servation the "index error" of the sextant. We have seen 
(p. 62) that when the two sextant mirrors are parallel, 
the index should read 0' 0". But it is seldom possible 
to adjust the instrument so that this condition will be satis- 
fied exactly ; nor would the adjustment remain perfect very 
long. A better plan is to determine by observation how 
much the reading differs from 0' 0", when the mirrors 
are parallel. This difference is the index error, and must 
be applied as a correction to all angles observed with the 
instrument. 

It is easy to make the mirrors parallel : we have merely 



THE SEXTANT 67 

to sight some distant well-defined terrestrial object like the 
gilt ball on the top of a flagpole (or the sea horizon, if aboard 
ship at sea), after clamping the index near 0. We shall 
then see in the telescope two images of the distant object; 
one by direct vision through the unsilvered part of the hori- 
zon glass, the other after reflection from both mirrors. By 
means of the tangent screw, the observer, with his eye at 
the telescope, can bring these two images together, so that 
they will appear as a single image. Then the mirrors will 
be parallel, and the vernier should read 0' 0". If it actually 
reads 8', for instance, instead of 0' 0", it means that the 
reading is 8' too large on account of index error ; and every 
angle measured with that sextant at that time will be 8' 
too large, and must be corrected by subtracting 8' from it. 

If, on the other hand, the reading is 8' "off the arc," 
when it should be 0', the instrument reads 8' too small, 
and any angle measured with it must be corrected by adding 
8' to it. 

For accurate determination of the index error (and it 
should be checked frequently), navigators prefer to observe 
the sun, or at night, a star. If a star is used, the process 
is the same as just described for a flagpole ball. But if 
the sun is used, a slightly different method is required. The 
sun, as seen in the telescope, shows a round disk of con- 
siderable size, and it is not possible to 
superpose the two images accurately. 
Therefore it is better to make them 
just touch, as shown in Fig. 15, when 
they are said to be "tangent" to each 
other. This must be done successively 
in two positions, AB and BA. In 

,, , ., ,, f, , .,, ,, FIG. 15. Index Error. 

other words, after the first "tangency 

has been observed, the tangent screw (B, fig. 13) is manipu- 
lated until the image A passes across B from top to bottom, 
and gives a new tangency in the second position. 

Each tangency will give a reading of the vernier. Unless 




68 NAVIGATION 

the sextant is greatly out of -adjustment, one of these read- 
ings will be off the arc, the other on the arc. If there were 
no index error, the off-arc and on-arc readings would be 
equal; if they differ, half the difference is the index error. 
If the off-arc reading is the larger, all altitudes measured 
with that sextant must be increased by the amount of the 
index error ; and if the on-arc reading is the larger, all such 
altitudes must be similarly diminished. 

The following is an example of an index error determina- 
tion: 

On-arc readings, Off-arc readings, 

31' 20" 33' 20" 

31 40 33 50 

30 50 34 

Means, 31' 17" 33' 43" 

The difference is 33' 43" - 31' 17" = 2' 26". Half the 
difference, or 1' 13", is the index error ; and because readings 
on the arc are the smaller, all angles read with this instru- 
ment must be increased by 1' 13", or, for ordinary purposes 
of navigation, by 1'. 

In addition to certain "adjusting screws" with which 
the index error can be reduced when it becomes unduly 
large, means are provided for three other sextant adjust- 
ments. These are : 

1. To make the index glass perpendicular to the frame of 
the instrument. 

2. To do the same with the horizon glass. 

3. To set the telescope parallel to the frame of the instru- 
ment. 

These adjustments are always completed by the maker 
before a sextant is sent out, nor does the navigator usually 
need to correct them himself. But it is important to know 
how to test them occasionally. Perpendicularity of the 
index glass can be examined by looking into the glass very 
obliquely with the index set near 0. It is then possible to 
see the inner edge of the sextant circle both by looking at 



THE SEXTANT 69 

it directly, past the edge of the index glass, and also by reflec- 
tion in the glass itself. The inner edge of the circle should 
form a continuous line when so examined, if the glass is 
perpendicular ; but if it is inclined, the line will appear broken, 
instead of continuous. 

Secondly, perpendicularity of the horizon glass can be 
tested at the same time the index error is determined by 
observing a star or a distant terrestrial point (p. 67). The 
index glass having been properly adjusted to perpendic- 
ularity, the two mirrors can never be made parallel by 
moving the index, unless the horizon glass is also properly 
perpendicular. Any existing lack of adjustment will there- 
fore betray itself in the index error determination, because 
the two images of the star or distant object will not be super- 
posed in any position of the index. 

Thirdly, the parallelism of the telescope to the frame of 
the instrument can usually be best tested with an ordinary 
pair of "calipers." 

Having thus described the sextant, its adjustments, and its 
use from the deck, we have still to explain how it can be used 
ashore. Sometimes it is necessary for the 5, 
navigator to make observations ashore, 
when it is not usually possible to see the 
horizon line (p. 61). Recourse must 
then be had to an "artificial horizon," 
which is simply an iron basin full of 
mercury covered with a glass roof. The 
mercury furnishes an almost perfectly 
horizontal mirror, and the glass roof 
prevents wind from ruffling the mercury 
surface, and thus destroying the mirror. p IG> 15. _ Artificial 
Figure 16 explains the principle of the Horizon, 

artificial horizon. HH is the mercury mirror, S the sun, 
and X the sextant. The observer aims the sextant telescope 
at the mercury where he can see a reflection of the sun. He 
then measures with the instrument the angle between a line 




70 



NAVIGATION 



drawn to the sun as seen reflected in the mercury and another 
line drawn to the actual sun in the sky. It can be shown 
by geometry that this measured angle will be just twice the 
real altitude of the sun, such as it would be if observed from 
the sea horizon. Therefore, in using the artificial horizon, 
it is merely necessary to divide the sextant angle by 2 to ob- 
tain the correct altitude of the sun. 

In observations of this kind two "suns" are seen at the 
same time in the telescope, just as is the case in index error 
observations (p. 67) ; whereas in observing from the sea 
horizon, the telescope shows only one solar image and the 
horizon line. When there are thus two solar images, they 
must be brought into tangency, just as we have already 
explained for index error (p. 67). When there is but one, 
it must be brought into tangency with the visible sea 
horizon line. 

But this altitude is not yet ready to be used in the further 
calculations for obtaining the position of the ship in latitude 

and longitude. Further pre- 
paratory corrections must be 
applied, in addition to the 
index error (p. 66), which is 
always the first correction to 
receive attention . These pre- 
paratory corrections are : 

1. "Dip" of the sea hori- 
zon, due to the elevation of 
the navigator on the ship's 
deck above the surface of the 
sea. Its cause is shown in 
Fig. 1 7 . C is the center of the 

Fio. 17. Dip of the Horizon. earth, K a point at sea level, 

and the navigator, elevated 

a distance OK above the sea. OZ is the direction of the ze- 
nith (p. 61), OS the direction of the sun, and OH a horizontal 
line from 0, OT is a line drawn through 0, and just touch- 




THE SEXTANT 71 

ing the sea surface at T'. Evidently OT will be the direc- 
tion of the sea horizon, where sky and sea seem to meet. 
Therefore, the altitude of the sun, as measured from the 
visible sea horizon, will be the angle SOT ; whereas the angle 
we require is the angle SOH, or the altitude of the sun 
above the true horizontal line OH. Therefore the angle 
HOT is a correction for dip which must be subtracted from 
all measured altitudes, and the amount of the correction 
depends on the height of the navigator's eye above the sea 
surface. 

2. "Refraction" is a bending of the light rays as they 
come down to us from the sun through the terrestrial atmos- 
phere. It always makes the sun seem higher in the sky 
than it really is, giving another subtractive correction for 
the observed altitude. The bending here involved is due 
to the passage of the sun's light rays through atmospheric 
strata of increasing density as the light approaches the 
earth's surface. 

3. "Parallax" is a small correction which must be added 
to the observed altitude of the sun. In strict theory, all astro- 
nomic observations are supposed to be made from the earth's 
center instead of its surface where the ship floats; and the 
small parallax correction allows for this minor theoretic 
point. In the case of star observations this correction is 
zero. 

4. " Semidiameter " is a correction depending on the 
choice by the navigator of a particular point on the sun's 
disk (p. 67) for observation. The sun's altitude, as used 
in the further calculations, should be the altitude of the sun's 
center ; but it is impossible to locate the center of the disk 
accurately in the telescope, so the navigator always observes 
the lowest point of the disk. This is called the "lower 
limb" of the sun. 

Beginners sometimes have difficulty in distinguishing 
the upper from the lower limb in the telescope. The best 
way to do this is to focus the telescope on some distant 



72 NAVIGATION 

object, and note whether it appears upside-down in the 
field of view. If so, the telescope is an "inverting" one, 
and the top of the sun must be observed, as it appears in 
the telescope, though it will really be the correct (or lower) 
limb, because of inversion by the telescope. When using the 
artificial horizon with an inverting telescope, the tangency 
must be made by bringing the bottom of the mercury image 
in contact with the top of the other image. The high-pow * 
ered telescopes supplied with good sextants are usually in- 
verting telescopes. 

Evidently the measured altitude, as it comes from the 
sextant, must be increased by the amount by which the sun's 
center is higher than the lower limb, and this is the sun's 
semidiameter. The index correction, together with the 
above four additional corrections, will fully prepare a meas- 
ured sextant altitude of the sun for further use in naviga- 
tional calculations. In the case of a star, which appears 
in the telescope as a point of light only, without any per^ 
ceptible disk, no semidiameter or parallax corrections are 
required; and in using the artificial horizon (p. 69), no 
correction for dip is necessary, either for the sun or a star. 

It is possible to arrange these various corrections in con- 
venient tables. Thus, in Table 6 (p. 247), we give a combi- 
nation of corrections 2 (refraction), 3 (parallax), and 4 (semi- 
diameter), to be used for observations of the sun's lower 
limb, and the same combination without the semidiameter 
and parallax l to be used for star observations. It will be 
noticed that the tabular corrections vary for different values 
of the observed altitude, which appears in the left-hand col- 
umn of the table. This variation comes mainly from the 
refraction part of the combined correction, for the refrac- 
tion is much greater when the sun or star is observed at a 
low altitude near the horizon than it is at a high altitude 
near the zenith. At the foot of the page is given a small 
supplementary correction depending on the date in the year. 
1 Which leaves refraction only. 



THE SEXTANT 73 

This small correction is not important in navigation, but is 
given here for the sake of completeness. It arises from the 
semidiameter part of the combined correction, for the an- 
nual orbit of the earth around the sun is of such a shape 
that the earth is nearer the sun in January than it is in July, 
which makes the sun appear bigger in January. And when the 
sun appears big, the semidiameter will of course be large too. 

Table 7 gives the dip of the sea horizon, the number in the 
left-hand column being the height (in feet) of the navigator's 
eye above sea level. This will be the height of the ship's 
deck, increased by the height of the man's eye above the 
deck. Unfortunately, the dip, as given in Table 7, at times 
varies considerably from the dip as it actually exists at the 
ship. The cause can be seen from Fig. 17 (p. 70), where 
it will be noticed that the line from the observer at to the 
sea horizon at T' passes very near the surface of the ocean. 
It is therefore entirely in the lowest strata of the terrestrial 
atmosphere, and there quite irregular refractions sometimes 
occur. These have been known to produce errors in the dip 
amounting to 10' or 20', and it is principally the existence 
of these unavoidable errors that makes it unnecessary to 
read the sextant closer than the nearest minute (p. 66), 
when observing from the deck. But when observing ashore 
with the artificial horizon, which has no dip, the navigator 
may, if he chooses, read seconds, especially if he intends to 
use in his further calculations the "mean" or average of 
a considerable number of observations. 

We shall now give an example of the complete correction 
of a sextant observation. Suppose the angle read from 
the sextant was 30 28', the index error (p. 68) 1', addi- 
tive, height of observer's eye 26 feet. We should then 
have : 

observed altitude, lower limb = 30 28' 

index correction = + 1' 

correction from Table 6 (p. 247) = + 14' 

correction from Table 7 (p. 247) = - 5' 

corrected altitude, for further use = 30 38' 



74 NAVIGATION. 

If the altitude had been observed ashore with an arti- 
ficial horizon, it might have been desirable to retain seconds. 
The calculation might then have been as follows : 

observed double altitude (see p. 70), lower limb = 63 0' 20" 

index correction (p. 68) = + 1 13 

corrected double altitude =63 1 33 

resulting altitude = 31 30 46 

correction from Table 6 (interpolated) = + 14 31 

corrected altitude, for further use =31 45 17 



CHAPTER VII 
THE NAUTICAL ALMANAC 

BEFORE beginning the further utilization of altitude ob- 
servations in our navigation calculations, it is necessary to 
understand the use of the Nautical Almanac. This is an 
annual publication, issued in two different editions by the 
Nautical Almanac Office, United States Naval Observatory. 
Copies can be obtained from the Superintendent of Docu- 
ments, Washington, D. C., or through any dealer in nautical 
supplies. Navigators do not need the larger edition, of which 
the title is "American Ephemeris and Nautical Almanac"; 
accordingly, all our references are made to the smaller edi- 
tion for the year 1917. Parts of certain pages from that 
edition are reprinted in the present volume for convenience 
of reference, and we shall give a somewhat detailed explana- 
tion of the almanac page 29 (our p. 76). 

Let us consider the date Monday, Dec. 17. We find for 
that date, and for every even hour (0*, 2*, 4*, 6*, etc.) of 
"Greenwich Mean Time" (abbreviated G. M. T. 1 ), two 
tabular numbers (p. 10) called "sun's declination" and 
"equation of time." 

To understand these it is necessary to bear in mind that 
the kind of time in ordinary use is "solar time," as kept by 
the sun. The "solar day" begins at "noon," called 0* in 
astronomic navigation, and it continues through twenty-four 
hours, without any confusing A.M. and P.M. In ordinary 
life the day begins twelve hours sooner, at midnight, and 
runs through two twelve-hour periods of A.M. and P.M. to 

1 The reader is requested to note carefully this abbreviation, as 
it will be used very frequently. 

75 



76 



NAVIGATION 



SUN, DECEMBER, 1917. From Nautical Almanac, p. 29 



G. M. T. 


SUN'S DEC- 
LINATION 


EQUATION 
OF TIME 


SUN'S DEC- 
LINATION 


EQUATION 
OP TIME 


SUN'S DEC- 
LINATION 


EQUATION 
OP TIME 




Monday 17 


Tuesday 25 


Saturday 29 


h 


/ 


m s 


1 


m s 


/ 


m s 





- 23 21.3 


+ 3 56.8 


- 23 24.7 


-0 1.6 


- 23 15.2 


- 1 59.7 


2 


23 21.5 


3 54.4 


23 24.6 


4.1 


23 14.9 


2 2.1 


4 


23 21.7 


3 51.9 


23 24.5 


6.5 


23 14.6 


2 4.6 


6 


23 21.9 


3 49.5 


23 24.4 


9.0 


23 14.3 


2 7.0 


8 


23 22.1 


3 47.0 


23 24.2 


11.5 


23 14.0 


2 9.4 


10 


23 22.2 


3 44.5 


23 24.1 


14.0 


23 13.7 


2 11.9 


12 


23 22.4 


3 42.1 


23 24.0 


16.5 


23 13.4 


2 14.3 


14 


23 22.6 


3 39.6 


23 23.8 


18.9 


23 13.1 


2 16.7 


16 


23 22.8 


3 37.1 


23 23.7 


21.4 


23 12.8 


2 19.1 


18 


23 22.9 


3 34.7 


23 23.5 


23.9 


23 12.5 


2 21.5 


20 


23 23.1 


3 32.2 


23 23.4 


26.4 


23 12.2 


2 24.0 


22 


23 23.2 


3 29.8 


23 23.2 


28.8 


23 11.9 


2 26.4 


H. D. 


0.1 


1.2 


0.1 


1.2 


0.1 


1.2 




Tuesday 18 


Wednesday 26 


Sunday 30 





- 23 23.4 


+ 3 27.3 


- 23 23.1 


- 31.3 


- 23 11.6 


- 2 28.8 


2 


23 23.6 


3 24.8 


23 22.9 


33.8 


23 11.3 


2 31.2 


4 


23 23.7 


3 22.3 


23 22.7 


36.3 


23 11.0 


2 33.6 


6 


23 23.8 


3 19.9 


23 22.5 


38.7 


23 10.6 


2 36.0 


8 


23 24.0 


3 17.4 


23 22.4 


41.2 


23 10.3 


2 38.4 


10 


23 24.1 


3 14.9 


23 22.2 


43.7 


23 10.0 


2 40.9 


12 


23 24.3 


3 12.5 


23 22.0 


46.2 


23 9.7 


2 43.3 


14 


23 24.4 


3 10.0 


23 21.8 


48.6 


23 9.3 


2 45.7 


16 


23 24.5 


3 7.5 


23 21.7 


51.1 


23 9.0 


2 48.1 


18 


23 24.6 


3 5.0 


23 21.5 


53.6 


23 8.6 


2 50.5 


20 


23 24.8 


3 2.6 


23 21.3 


56.0 


23 8.3 


2 52.9 


22 


23 24.9 


3 0.1 


23 21.1 


58.5 


23 7.9 


2 55.3 


H. D. 


0.1 


1.2 


0.1 


1.2 


0.2 


1.2 




Wednesday 19 


Thursday 27 


Monday 31 





- 23 25.0 


+ 2 57.6 


- 23 20.9 


- 1 0.9 


- 23 7.6 


- 2 57.7 


2 


23 25.1 


2 55.1 


23 20.7 


1 3.4 


23 7.2 


3 0.1 


4 


23 25.2 


2 52.6 


23 20.5 


1 5.9 


23 6.9 


3 2.4 


6 


23 25.3 


2 50.2 


23 20.3 


1 8.3 


23 6.5 


3 4.8 


8 


23 25.4 


2 47.7 


23 20.1 


1 10.8 


23 6.1 


3 7.2 


10 


23 25.5 


2 45.2 


23 19.8 


1 13.2 


23 5.8 


3 9.6 


12 


23 25.6 


2 42.7 


23 19.6 


1 15.7 


23 5.4 


3 12.0 


14 


23 25.7 


2 40.2 


23 19.4 


1 18.1 


23 5.0 


3 14.4 


16 


23 25.8 


2 37.8 


23 19.2 


1 20.6 


23 4.6 


3 16.7 


18 


23 25.9 


2 35.3 


23 19.0 


1 23.1 


23 4.3 


3 19.1 


20 


23 26.0 


2 32.8 


23 18.7 


1 25.5 


23 3.9 


3 21.5 


22 


23 26.1 


2 30.3 


23 18.5 


1 28.0 


- 23 3.5 


- 3 23.9 


H. D. 


0.0 


1.2 


0.1 


1.2 


0.2 


1.2 




Thursday 20 


Friday 28 







- 23 26.1 


+ 2 27.8 


- 23 18.3 


^ 1 30.4 




2 


23 26.2 


2 25.3 


23 18.0 


1 32.9 




4 


23 26.3 


2 22.8 


23 17.8 


1 35.3 




6 


23 26.3 


2 20.4 


23 17.5 


1 37.8 




8 


23 26.4 


2 17.9 


23 17.3 


1 40.2 


SEMIDIAMETER 


10 


23 26.5 


2 15.4 


23 17.0 


1 42.6 




12 


23 26.5 


2 12.9 


23 16.8 


1 45.1 






14 


23 26.6 


2 10.4 


23 16.5 


1 47.5 


Dec. 1 


16'26 


16 


23 26.6 


2 7.9 


23 16.3 


1 50.0 


11 


16'28 


18 


23 26.7 


2 5.4 


23 16.0 


1 52.4 


21 


16'29 


20 


23 26.7 


2 2.9 


23 15.7 


1 54.8 


31 


16'30 


22 


- 23 26.8 


+ 2 0.4 


- 23 15.4 


- 1 57.3 




H. D. 


0.0 


1.2 


0.1 


1.2 





NOTE. The Equation of Time is to be applied to the G. M. T. in accordance with 
the sign as given. 



THE NAUTICAL ALMANAC 77 

the following midnight; but this "civil day," as it is called, 
does not for the moment concern us. 

Solar time, as kept by the visible sun, is a very incon- 
venient kind of time, because there are certain peculiarities 
in the astronomic motion of the earth which make these 
solar days of unequal length. They are called "apparent 
solar days" and the corresponding kind of time is "apparent 
solar time." 

To avoid the above inconvenience, an imaginary "mean 
sun" and a "mean solar day" have been invented. The 
mean sun conforms as nearly as possible to the average per- 
formance of the visible sun, and the length of the mean 
solar day is the average of all the apparent solar days through- 
out the year. The corresponding kind of time, kept by the 
mean sun, is "mean solar time" ; and this is the kind of time 
recorded by all our watches and marine chronometers (p. 6). 

The difference between these two kinds of solar time varies 
on different dates, and even at different hours on the same 
date. It is this difference which is called the "equation of 
time " and which is one of the tabular numbers in the nautical 
almanac page 29 (our p. 76). 

This equation of time is of great importance in navigation, 
and it is easy to see how page 29 of the almanac may be used 
to find it. Suppose, for instance, we wish to know what the 
equation is on Dec. 17, 1917, on board ship, when the ship's 
chronometer indicates on its face 3 P.M., civil time, or (which 
is the same thing) 3*, astronomical time (p. 75). Ship's 
chronometers are always set to Greenwich mean time, so 
that 3 A by the chronometer signifies that the time at Green- 
wich was 3\ 

We then look in the almanac page 29 (our p. 76), and find 
that the equation was + 3 W 54*.4 at 2 h , G. M. T., and 
+ 3 m 5P.9 at 4*, G. M. T. Its value at 3* must be half- 
way between these two, or + 3 m 53*. 15. This we would 
call + 3 m 53*.2, so as to avoid the use of hundredths of 
seconds, which do not need attention in navigation. And 



78 NAVIGATION 

since the equation is merely the difference between the 
two kinds of solar time, the + sign means that it must be 
added to G. M. T., to obtain Greenwich apparent time, in 
accordance with the "Note" at the foot of the almanac 
page 29. Consequently, the G. M. T. by chronometer having 
been 3 h O m 0*, the Greenwich apparent time at the same in- 
stant was 3* 0" 1 + 3 m 53 f .2 = 3* 3 OT 53*.2. 

It will be noticed that the process we have here used for 
obtaining the equation from the almanac is merely an inter- 
polation (see p. 12). Let us, as another example, find the 
equation for Sunday, Dec. 30, at 10* 26 m A.M., civil time by 
chronometer, and we have purposely here retained the 
civil method of reckoning time to make certain that the 
reader understands the difference between civil and astro- 
nomic (or navigation) time. The given time is 10* 26 m A.M., 
civil time, Dec. 30. But the astronomic Dec. 30 does not 
begin until noon (p. 75), so that it is not yet Dec. 30 by 
astronomic reckoning. By that reckoning it is really only 
22 h 2Q m on Dec. 29. In other words, when the civil time is 
P.M., as in the first example, the astronomic time is the same 
as the civil time. But when the civil time is A.M., as in the 
present example, the astronomic time is found by adding 
12* to the civil time, and deducting 1 from the date. These 
complications emphasize the advantage of the astronomic 
count, which avoids A.M. and P.M. altogether. 

We now have from the almanac (p. 76) : 

equation of time, Dec. 29, 22 A , G. M. T. = - 2 m 2Q'A, 
equation of time, Dec. 30, A , G. M. T. = - 2 m 28'.8 ; 

and the numbers in this example have been purposely so 
chosen that the above two tabular values of the equation 
(between which the required value falls) come from different 
dates in the almanac. This creates no confusion, for these 
two values of the equation are really consecutive tabular 
numbers, just as much as if they occurred on a single date. 
The difference between the two values of the equation is 



THE NAUTICAL ALMANAC 79 

2*.4; and as this difference corresponds to 2 h in the left- 
hand (or argument) column, it follows that the difference 
for l h is here P.2. This is the change of the equation per 
hour of time; it is called the "hourly difference" (abbre- 
viated H. D.) and is printed in the almanac at the foot of 
each daily column. 

Now we want the equation for Dec. 29, 22 A 26 TO , by the 
chronometer. The 26 must next be changed into a decimal 
fraction of an hour. 26 m = ff of an hour = A .43. So the 
time for which we want the equation becomes Dec. 29, 
22 A .43. The H. D. being P.2, the change in OM3 will be 
1'.2 X 0.43 = 0*.5. The almanac shows that at 22 A the equa- 
tion was 2 OT 26*. 4, and was increasing numerically. There- 
fore, at 22 A .43, it was 2 m 26 8 .4 + 0'.5 = 2 m 26'.9. And this 
number has the minus sign. Therefore, the G. M. T. being 
Dec. 29, 22* 26 m , the Greenwich apparent time at the same 
instant will be Dec. 29, 22* 26 m - 2 m 26*.9 = Dec. 29, 
22* 23 33M. 

Most of these minor interpolation calculations, which are 
here set forth in great detail for the benefit of the beginner, 
can be made with sufficient accuracy by a skilled navigator 
mentally. 

In the foregoing two examples we have assumed that the 
chronometer was right, but these instruments practically 
never run quite correctly. Therefore, before leaving port, 
navigators always have their chronometers "rated" by a 
chronometer expert; and when the instrument is returned 
to the ship just before sailing, a "rate card" (or "rate paper") 
always comes with it. Let us suppose that in the present 
example this card stated that the chronometer was slow 
8 m 22'. 5 x on Dec. 20, at noon, and was "losing" 2 1 8 .8 daily. 
The 8 ro 22*. 5 would then be the "chronometer error" on 
Dec. 20 ; and the 1*.8 would be its "daily rate." 

1 This number is here purposely chosen much larger than would 
ever occur in practice. 

2 The opposite kind of "rate" is called "gaining." 



80 NAVIGATION ., 

From Dec. 20, noon, to Dec. 30, 10* 26 TO A.M. is an interval 
of 9 days 22 hours 26 minutes. This interval must now be 
reduced to a decimal of a day. 26 m = $ of an hour = A .43. 
The interval is therefore 9* 22 A .43. 

But 22 A .43 = 2 -$* days = O tf .93. Therefore, in days, the 
interval is 9 a .93. This transformation of hours and minutes 
into decimals of a day can be accomplished with less trouble by 
means of our Table 8 (p. 248). 

Having a losing rate of P.8 daily, the chronometer lost 
1'.8 X 9.93 = 17*.9 in the interval of 9.93 days. And as it was 
already slow 8 m 22 s . 5 on Dec. 20, it was slow 8 m 22*.5 + 17*.9 
= 8 m 40 s . 4 at the time for which the equation is. required. 

Now the equation was required for Dec. 29, 22* 26 OT by the 
chronometer; and that instrument being slow 8 TO 40*.4, the 
correct G. M. T. was : Dec. 29, 22 h 26 m + 8 m 40*.4 = Dec. 29, 
22* 34 40* .4. Turned into a decimal fraction of an hour, 
this becomes Dec. 29, 22 A .58, instead of 22 h A3, as we found 
before, when the chronometer error was omitted from the 
calculation. The H. D. is 1*.2, as before, and the change 
in ' A .58 = K2 X 0.58 = 8 .7. Therefore, at 22 A .58 the 
equation is 2 m 26 8 .4 + 0*.7 = 2 m 27M. This still has the 
minus sign, so that the correct Greenwich apparent time 
becomes Dec. 29, 22* 34" 1 40'.4 - 2 m 27M = 22 A 32 m 13 S .3. 

All the above calculations have been carried out here with 
unnecessary accuracy. There would be no harm if the result 
were in error by a few tenths of a second ; and it is this cir- 
cumstance that makes it possible to perform these inter- 
polations largely mentally. 

In the foregoing examples no account was taken of the 
ship's location on the ocean; yet this location may have an 
indirect influence on the calculations. To understand this, 
we must consider for a moment the time-differences which 
exist between different places on the earth. The sun rises in 
the east and travels across the sky toward the west ; so that 
if we consider two places like Greenwich, England, and New 
York, for instance, the sun, because of this motion from east 



THE NAUTICAL ALMANAC 81 

to west, will pass Greenwich first. Consequently, when it is 
noon in New York, it has already been noon in Greenwich, 
and is afternoon there. Greenwich time is therefore always 
later than New York time. The same is true of any other 
two places ; there is always a time-difference between them, 
and the easterly place has the later or "faster" time. 

The amount of such time-difference of course depends 
on the relative location of the two places, and the relation is 
such that 15 of longitude-difference corresponds exactly 
to l h of time-difference. Thus Sandy Hook, which is in 
longitude 73 50' west of Greenwich, has a time-difference 
from Greenwich of 4* 55 m 20*. This conversion of longitude 
into time-difference is best accomplished by means of our 
Table 9 (p. 249). According to that table : 

73 = 4* 52* 

50' 3 20 

73 50'' = 4* 55" 20 

The indirect influence of such time-differences upon the 
use of the almanac is that they may at times, especially 
when they are large, make the Greenwich date of the ob- 
servation different from the date on board. Thus a vessel 
off Manila Bay, in longitude 120 east of Greenwich, would 
have her local time 8 ft (120) later than Greenwich time. If 
a sextant observation was made on board at 4 P.M., civil 
time, on a Thursday, the chronometer would indicate S h , 
and it would be 8 A.M. on Thursday, because Greenwich is 
8 h earlier than the ship. This 8 A.M. would really be 20* of 
the preceding Wednesday by astronomic time, and so the 
almanac date used would be one day earlier than the date 
of the observation. The chronometer will always give the 
right Greenwich time, but the navigator must be very care- 
ful to interpolate the almanac numbers on the right date. 

We have now learned how to ascertain the equation of 
time from the almanac, and how to use it for transforming 
G. M. T. into Greenwich apparent time. The contrary 
transformation, from Greenwich apparent time to G- M. T., 



82 NAVIGATION 

can be made by applying the equation in the opposite way : 
subtracting when it has the + sign in the almanac, and add- 
ing when it has the sign. 

The great importance of these time transformations comes 
from the fact that sextant observations must necessarily be 
made upon the visible sun. When they are made for the 
purpose of calculating the local time on board, this local 
time will therefore necessarily be local apparent solar time, as 
kept by the visible sun. At the instant of the observation 
(p. 6), the chronometer face (corrected for error and rate) 
tells us the G. M. T. If this is turned into Greenwich ap- 
parent time by applying the equation, we have only to com- 
pare the Greenwich and the ship's apparent times to get 
the time-difference between the ship and Greenwich. This 
time-difference can then be turned into degrees and minutes, 
and will be the ship's longitude. Examples of this calcu- 
lation will be given in detail (p. 99). It is also worth 
noting here that the time-difference between any two places 
is precisely the same, quite irrespective of the kind of time 
in which it is counted. 

To complete our explanation of the almanac page 29 (our 
p. 76), it remains to give an example of a calculation of the 
sun's declination. This is an angle in degrees and minutes, 
and it is interpolated just like the equation by the aid of 
its H. D. Thus, for Dec. 29, 22*.58 (p. 80) the declination 
is obtained thus : 

Dec. 29, 22*, declination = 23 1 1 '.9 

H.D. (O'.l) x 0*.58 = 0.1, declination decreasing ; 

by subtraction, at 22*.58, dec. = 23 11 '.8, 

and according to the almanac, this declination must be given 
the minus sign. When the sign should be +, that fact is 
indicated in the almanac. The use of the declination will 
be explained later; the accuracy required in the interpo- 
lation of it is not so great as we have used here, for the 
nearest minute suffices in practically all navigation work. 
In addition to the sun's declination, navigators require 



THE NAUTICAL ALMANAC 



83 



in their further calculations another number called the sun's 
"right ascension" (abbreviated, R. A.). This is obtained 
from pages like the almanac page 3 (reprinted in part below). 
It is always the R. A. of the "mean sun" that we need, 
and the almanac gives it for Greenwich mean noon of each 
day in the year. When needed in our further calcula- 
tions, it is of course always required for the exact moment 
when a sextant observation was made. In fact, this state- 
ment applies also to the equation of time and declination. 
They must always be interpolated from the almanac for the 
moment when the navigator actually observed the sun ; and 

SUN, 1917. From Nautical Almanac, p. 3 



DAY 

OF 

MONTH 


RIGHT ASCENSION OF THE MEAN SUN AT GREENWICH MEAN NOON 


July 


August 


September 


October 


November 


December 




i m a 


h m a 


h m a 


h m s 


h m a 


h m s 


1 


6 35 52.2 


8 38 5.5 


10 40 18.7 


12 38 35.3 


14 40 48.4 


16 39 5.1 


2 


6 39 48.8 


8 42 2.0 


10 44 15.2 


12 42 31.8 


14 44 45.0 


16 43 1.7 


3 


6 43 45.3 


8 45 58.6 


10 48 11.8 


12 46 28.4 


14 48 41.5 


16 46 58.2 


4 


6 47 41.9 


8 49 55.1 


10 52 8.3 


12 50 24.9 


14 52 38.1 


16 50 54.8 


5 


6 51 38.4 


8 53 51.7 


10 56 4.9 


12 54 21.5 


14 56 34.6 


16 54 51.3 


6 


6 55 35.0 


8 57 48.2 


11 1.4 


12 58 18.0 


15 31.2 


16 58 47.9 


7 


6 59 31.6 


9 1 44.8 


11 3 58.0 


13 2 14.6 


15 4 27.8 


17 2 44.5 


8 


7 3 28.1 


9 5 41.4 


11 7 54.5 


13 6 11.1 


15 8 24.3 


17 6 41.0 


9 


7 7 24.7 


9 9 37.9 


11 11 51.1 


13 10 7.7 


15 12 20.9 


17 10 37.6 


10 


7 11 21.2 


9 13 34.5 


11 15 47.6 


13 14 4.2 


15 16 17.4 


17 14 34.1 


11 


7 15 17.8 


9 17 31.0 


11 19 44.2 


13 18 0.8 


15 20 14.0 


17 18 30.7 


12 


7 19 14.3 


9 21 27.6 


11 23 40.8 


13 21 57.3 


15 24 10.5 


17 22 27.2 


13 


7 23 10.9 


9 25 24.1 


11 27 37.3 


13 25 53.9 


15 28 7.1 


17 26 23.8 


14 


7 27 7.4 


9 29 20.7 


11 31 33.9 


13 29 50.4 


15 32 3.6 


17 30 20.4 


15 


7 31 4.0 


9 33 17.2 


11 35 30.4 


13 33 47.0 


15 36 0.2 


17 34 16.9 


16 


7 35 0.6 


9 37 13.8 


11 39 27.0 


13 37 43.6 


15 39 56.8 


17 38 13.5 


17 


7 38 57.1 


9 41 10.4 


11 43 23.5 


13 41 40.1 


15 43 53.3 


17 42 10.0 


18 


7 42 53.7 


9 45 6.9 


11 47 20.1 


13 45 36.7 


15 47 49.9 


17 46 6.6 


19 


7 46 50.2 


9 49 3.5 


11 51 16.6 


13 49 33.2 


15 51 46.4 


17 50 3.2 


20 


7 50 46.8 


9 53 0.0 


11 55 13.2 


13 53 29.8 


15 55 43.0 


17 53 59.7 


21 


7 54 43.4 


9 56 56.6 


11 59 9.7 


13 57 26.3 


15 59 39.5 


17 57 56.3 


22 


7 58 39.9 


10 53.1 


12 3 6.3 


14 1 22.9 


16 3 36.1 


18 1 52.8 


23 


8 2 36.5 


10 4 49.7 


12 7 2.8 


14 5 19.4 


16 7 32.6 


18 5 49.4 


24 


8 6 33.0 


10 8 46.2 


12 10 59.4 


14 9 16.0 


16 11 29.2 


18 9 46.0 


25 


8 10 29.6 


10 12 42.8 


12 14 55.9 


14 13 12.5 


16 15 25.8 


18 13 42.5 


26 


8 14 26.1 


10 16 39.4 


12 18 52.5 


14 17 9.1 


16 19 22.3 


18 17 39.1 


27 


8 18 22.7 


10 20 35.9 


12 22 49.0 


14 21 5.6 


16 23 18.9 


18 21 35.6 


28 


8 22 19.2 


10 24 32.4 


12 26 45.6 


14 25 2.2 


16 27 15.4 


18 25 32.2 


29 


8 26 15.8 


10 28 29.0 


12 30 42.2 


14 28 58.8 


16 31 12.0 


18 29 28.7 


30 


8 30 12.4 


10 32 25.6 


12 34 38.7 


14 32 55.3 


16 35 8.6 


18 33 25.3 


31 


8 34 8.9 


10 36 22.1 


12 38 35.3 


14 36 51.9 


16 39 5.1 


18 37 21.9 



84 



NAVIGATION 



CORRECTION TO BE ADDED TO R. A. M. S. AT G. M. N. FOR 

TIME PAST NOON 
From Nautical Almanac, p. 3, Continued 



TIME 


Qtn 


6" 1 


12- 


18 m 


Mm 


SO" 1 


36 m 


42m 


" 


TIME 


h 
12 
13 
14 
15 


m s 
1 58.3 
2 8.1 
2 18.0 
2 27.8 


m a 
1 59.3 
2 9.1 
2 19.0 
2 28.8 


m s 
2 0.2 
2 10.1 
2 20.0 
2 29.8 


m s 
2 1.2 
2 11.1 
2 20.9 
2 30.8 


m s 
2 2.2 
2 12.1 
2 21.9 
2 31.8 


m s 
2 3.2 
2 13.1 
2 22.9 
2 32.8 


m s 
2 4.2 
2 14.0 
2 23.9 
2 33.8 


m s 
2 5.2 
2 15.0 
2 24.9 
2 34.7 


m s 
2 6.2 
2 16.0 
2 25.9 
2 35.7 


h 
12 
13 
14 

15 


16 
17 
18 
19 


2 37.7 
2 47.6 
2 57.4 
3 7.3 


2 38.7 
2 48.5 
2 58.4 
3 8.3 


2 39.7 
2 49.5 
2 59.4 
3 9.2 


2 40.7 
2 50.5 
3 0.4 
3 10.2 


2 41.6 
2 51.5 
3 1.4 
3 11.2 


2 42.6 
2 52.5 
3 2.3 
3 12.2 


2 43.6 
2 53.5 
3 3.3 
3 13.2 


2 44.6 
2 54.5 
3 4.3 
3 14.2 


2 45.6 
2 55.4 
3 5.3 
3 15.2 


16 
17 
18 
19 


20 
21 
22 
23 


3 17.1 
3 27.0 
3 36.8 
3 46.7 


3 18.1 
3 28.0 
3 37.8 
3 47.7 


3 19.1 
3 29.0 
3 38.8 
3 48.7 


3 20.1 
3 29.9 
3 39.8 
3 49.7 


3 21.1 
3 30.9 
3 40.8 
3 50.6 


3 22.1 
3 31.9 
3 41.8 
3 51.6 


3 23.0 
3 32.9 
3 42.8 
3 52.6 


3 24.0 
3 33.9 
3 43.7 
3 53.6 


3 25.0 
3 34.9 
3 44.7 
3 54.6 


20 
21 
22 
23 



the Greenwich time of this event is of course always taken 
from the chronometer (duly corrected for error and rate). 

Thus, if the R. A. of the mean sun is required for Dec. 29, 
22* 34 m 40.4, G. M. T. (p. 80), we find from the almanac 
page 3 (our p. 83) that the R. A. of the mean sun at Green- 
wich mean noon is 18* 29 m 28*.7. x This, according to the sup- 
plementary table quoted above from page 3, must be increased 
by a correction for "time past noon." In this case the time 
past noon is 22* 34 m 40*.4. The tabular correction for 22* 30 
is 3 m 41'.8, and for 22* 36 m it is 3 m 42'.8. Ours falls between 
these two, and an interpolation makes the correction 3 m 42*.6. 
Consequently, the R. A. of the mean sun for Dec. 29, 22* 
34* 40.4, G. M. T. is 18* 29" 28'.7 + 3 m 42'.6 = 18* 33 m 11*.3. 

It will be noticed that the small supplementary table 
(quoted above from almanac page 3) only runs from 12* to 24*. 
The other half of the table, from 0* to 12*, is printed on the 
opposite page 2 of the almanac. There is also another 
longer table, printed near the end of the almanac, and there 
called Table III, from which the supplementary correction 
can be taken without the necessity of interpolation. 

It is not absolutely essential that the navigator learn what 

1 Right ascensions are always thus measured in hours, minutes, 
and seconds, like time, and they are counted from 0* to 24*. 



THE NAUTICAL ALMANAC 85 

the words "right ascension" and "declination" really mean. 
But for the benefit of those who are curious in such matters 
we may state that these numbers locate the position of the 
sun (or of a star) on the sky. The sky is a great globe, called 
by astronomers the "celestial sphere," and all heavenly 
bodies are located upon it precisely as points on the earth 
are there located by their latitudes and longitudes (p. 3). 
There is a "celestial equator" with two "celestial poles," 
corresponding accurately to the terrestrial equator and poles. 
Declination then corresponds exactly to latitude on the earth, 
and so it measures the distance of a heavenly body from the 
celestial equator. When the body is north of the celestial 
equator, the declination is called +. 

Right ascension similarly corresponds to longitude ; and for 
the beginning point of right ascensions on the sky there is a 
"celestial Greenwich," which is called the "vernal equinox." 

After this brief digression into astronomy, we return to 
our subject. We have seen (p. 82) that observations of 
the sun will tell us only apparent solar time, because it is 
only, the visible sun that we can observe. If the observations 
are made upon a star, the kind of time is different from any 
so far mentioned. It is called "sidereal time," or star time. 

It is always possible to change mean solar time into sidereal 
time, and vice versa, by a simple process of calculation ; but 
the only change of this kind required in navigation is the 
transformation of G. M. T. into Greenwich sidereal time. 
To make this transformation, we have only to take from the 
almanac, for the given G. M. T., the R. A. of the mean sun, 
and then to add it to the given G. M. T. 

Thus, to find the Greenwich sidereal time corresponding 
to Dec. 29, 22* 34 m 40*.4, G. M. T., we have already found 
(p. 84) that the R. A. of the mean sun = 18* 33" 11*. 3 

To this must be added the given G. M. T. = 22 34 40.4 
Sum .= corresponding Greenwich sidereal time = 17* 1 7 m 51*.7 

1 The number of hours was here really 41* : but whenever it is 
larger than 24*, we must drop or reject 24*. 



CHAPTER VIII 
OLDER NAVIGATION METHODS 

WE shall now explain in detail certain standard methods 
of determining a ship's latitude and longitude by means of 
sextant observations. An understanding of these methods 
is essential to a proper comprehension of the newer naviga- 
tional processes to be described later ; and the older methods 
are in fact still very widely used at sea, although most re- 
cent authorities believe they should be rejected in favor of 
the newer procedure. 

The simplest of these older processes, and the one most 
frequently employed, is the determination of the ship's 
latitude by a noon or "meridian" observation ("noon- 
sight") of the sun's altitude (p. 61). Now the sun is 
higher in the sky at noon than it is at any other time during 
the day ; and so it is possible to get the noon-sight by be- 
ginning to observe the sun with the sextant a few minutes 
before noon, and continuing the observation as long as the 
sun's altitude is increasing. The moment it begins to 
diminish, or the sun to "dip," as sailors say, the observation 
should be terminated, and the vernier read. 

The altitude thus observed will be an altitude of the lower 
limb (p. 71) ; and before it is used further it must be fully 
corrected for index error ; for refraction parallax and semi- 
diameter ; and for dip ; all as in the example on p. 73, 
where the observed altitude was 30 28', and we found the 
corrected altitude to be 30 38'. 

Next, the sun's declination must be taken from the al- 
manac, being interpolated for the Greenwich time of the 

86 



87 

observation, as in the example on p. 82, where we found 
the declination to be - 23 12' on Dec. 29, at 22* 34 m 40'.4, 
G. M.T. We shall suppose the above altitude 30*28' to 
have been observed at the Greenwich time stated, so as to 
make use of the results of our former calculated examples. 
Nor is there any inconsistency in supposing a noon observa- 
tion to have been made at 22* 34 m 40*.4. For the noon 
observation is made when it is noon on board ship, while 
the 22* 34 m 40v4 is the G. M. T. at the same moment. 
The difference is simply the time-difference (p. 80) between 
Greenwich and the ship. 

The calculation of the ship's latitude is now made by the 
following formula : 

Latitude = 90 + Declination Altitude. 

In this formula, the plus sign signifies that the declination 
must be added; and the minus sign signifies that the altitude 
must be subtracted. Furthermore, it is most important to 
remember that if the declination is itself a "minus declina- 
tion," as in this example, the addition of it according to the 
formula is really a subtraction. Or, in other words, and in 
general, whenever a formula calls for an addition, and the 
number to be added is a minus number, then that number 
must be subtracted instead of added. And similarly, if the 
formula calls for a subtraction, and the number to be sub- 
tracted is a minus number, then that number must be added 
instead of subtracted. Two minus signs neutralize each other. 

In the present case we have, omitting seconds : 

90 0' 

declination =-23 12 

90 + declination = 66 48 
altitude = 30 38 

latitude = 36 10 

In considering this result it is of interest to inquire where 
this observation really locates the ship. Now we have not 
yet stated what the date was, on board, when the observa- 



88 NAVIGATION 

tion was made ; but we have given the G. M. T. as Dec. 29, 
22* 34 m 40* .4. The noon-sight was taken, as a matter of 
fact, afc noon on Dec. 30, or at the moment when the date 
Dec. 30 commenced by astronomic reckoning. Therefore 
the ship's time was later than the Greenwich time by about 
1* 25" ; or 21 15', allowing 15 to 1* (p. 81) ; and the ship 
was (approximately) in 21 15' east longitude from Greenwich. 
This, together with the latitude 36 10', locates the ship in 
the Mediterranean, south of Greece, and west of Candia. 

Although we have thus apparently located the ship com- 
pletely in latitude and longitude from a single noon-sight, 
it must not be supposed that we have really accomplished 
this. The noon-sight is only suitable for ascertaining the 
ship's latitude ; the longitude is determined so inaccurately 
as to be practically useless. The reason for this is that 
near noon the sun changes its altitude very slowly, because 
it is then near the turning-point where its upward morning 
motion is about to become a downward afternoon motion. 
For the sun's daily motion in the sky is upward in the morn- 
ing and downward in the afternoon. Near noon it runs 
along horizontally, or very nearly so, for several minutes, 
so that its altitude change is insignificant during that time. 

It follows from this temporary invariability of altitude 
that we cannot determine the exact moment when noon 
occurs by observing altitude changes with the sextant. But 
the latitude determination is not affected; because, for 
the latitude, we only need to know the noon altitude. And 
if we happen to measure it a little too soon or too late, on 
account of the difficulty of fixing the moment of noon, no 
harm will result, because the altitude very near noon is the 
same as it is at noon precisely, 'as we have just seen. 

It is, in general, practically impossible to determine both 
latitude and longitude from a single observation. To deter- 
mine two unknown things, at least two different observations 
must be made. Nor can any skillful method of planning 
the observation overcome this fundamental circumstance. 



OLDER NAVIGATION METHODS 89 

Returning now to our latitude formula (p. 87), it is 
necessary to modify it somewhat in case we happen to be in 
the tropics, where the sun may pass between the zenith and 
the celestial pole. Even in temperate latitudes a celestial 
body may do this, if we happen to observe a star instead of 
the sun. In such a case, if the ship is in the northern 
hemisphere, the navigator will observe the sun's altitude 
toward the north at noon instead of toward the south, as 
usual. Furthermore, in very high northern latitudes, the 
"midnight sun," as it is called, can be observed toward the 
north, and below the celestial pole. This is the minimum 
altitude during the day, instead of the maximum ; but it is 
usable for a latitude determination. Such an observation is 
called a "lower transit" ; and it can often be observed in the 
case of stars in temperate latitudes. 

If we now remember to call northerly latitudes and 
declinations plus, and southerly ones minus, we have the 
following complete set of formulas for the present problem, 
including observations in both hemispheres. These formulas 
are so arranged that we can easily choose the right formula, 
by having regard to the + and signs. But the right 
formula once chosen, the latitude is calculated without 
marking declinations with either the + or sign. 

if lat. greater than dee., lat. = 90 + dec. alt. (1) 
if dec. greater than lat., lat. = dec. + alt. - 90 (2) 



lat. 1 and 
dee. both + 
or both 



if lower transit, lat. = 90 + alt. - dec. (3) 



lat. and dec., 1 lat = ^ _ alt _ dec (4) 

one +, one j 

We shall now give some more examples ; and to enable 
the reader to follow star observations correctly we reprint 
part of the upper halves of pages 94 and 95 (our pp. 91, 92) 
of the Nautical Almanac. These contain the right ascensions 
and declinations (p. 85) of a quantity of bright stars for 
various dates in the year. These numbers are correct for the 
moment of "upper transit," which is the moment when these 
1 Latitude and declination are abbreviated lat. and dec. 



90 NAVIGATION 

stars attain their maximum altitudes. This event cannot 
be called a noon-sight in the case of a star ; but it is observable 
in a manner perfectly similar to a solar noon-sight. 

These stellar right ascensions and declinations change 
so slowly that it is unnecessary to use interpolation when 
taking them from the almanac pages. 

Proceeding now to our examples, suppose that on shore, 
at Sandy Hook Light, approximate latitude and longitude 
40 28' N., 74 0' W., on Monday, Dec. 17, 1917, at noon, the 
double altitude of the sun's lower limb was observed with a 
sextant and artificial horizon, and found to be 51 48'. The 
index correction required by the sextant was + 4' ; and the 
G. M. T. by chronometer was 4* 56 TO at the moment the 
observation was made. Find the latitude. We have : 

Observed double altitude 51 48' (1) 

Index correction + 4 (2) 

Adding (1) and (2) gives corrected double altitude 51 52' (3) 

Halving (3) gives observed altitude 25 56 (4) 

Correction from Table 6 1 (p. 247) + 14^ (5) 

Adding (4) and (5) gives fully corrected altitude 26 10' (6) 

Now use formula (4) (p. 89) because latitude is + 

and declination is - . Write 90 (7) 

Subtracting (6) from (7) gives 90 - corrected altitude . . 63 50 (8) 
Interpolate declination from almanac (p. 76). This 

gives declination 23 22 (9) 

Subtracting (9) from (8) gives for the latitude 40 28 (10) 

With regard to the foregoing example it is worth remark- 
ing that if there had been no available chronometer set to 
Greenwich time, it would still have been possible to calculate 
the observation. For the known approximate longitude, 
even if only a dead-reckoning (p. 5) longitude, would be 
quite accurate enough to make possible the interpolation of 
the declination from the almanac. And in the present 
example, the chronometer was only used in getting the 
declination printed in line (9) above. 

1 Dip correction from Table 7 not needed because the artificial 
horizon was used. 



OLDER NAVIGATION METHODS 



91 



APPARENT PLACES OF STARS, 1917 

From Nautical Almanac, p. 94 
FOR THE UPPER TRANSIT AT GREENWICH 







RIGHT ASCENSION 


Mr* 


CONSTELLA- 




^ 


^ 


1-4 




TH 


M 


' 


,_, 


rt 


<N 

CO 


IN t/ 


TION NAME 




q 


>> 


O 


S. 


. 


j 

















3 




S 


1 


3 

1-3 


8 

< 


I 


1 


1 


1 


1 






h in 


s 


s 


8 


S 


8 


s 


s 


s 


s 


8 


1 


<* Androm. 


4 


6.3 


6.4 


7.4 


8.4 


9.4 


10.0 


10.3 


10.3 


10.0 


9.6 


2 


ft Cassiop. 


4 


44.8 


44.4 


45.7 


47.3 


48.7 


49.7 


50.1 


49.9 


49.3 


48.4 


3 


0Ceti 


039 


26.5 


26.3 


27.0 


28.0 


28.9 


29.7 


30.0 


30.1 


29.8 


29.5 


4 


& Cassiop. 


1 20 


23.9 


22.3 


23.5 


25.1 


26.7 


28.1 


28.9 


29.2 


29.0 


28.2 


5 


Urs. Min. 


1 29 


89.0 


22.9 


45.5 


77.6 


112.8 


142.4 


161.2 


166.4 


155.3 


129.0 


6 


a Eridani 


1 34 


39.1 


36.8 


37.6 


38.8 


40.3 


41.5 


42.3 


42.4 


41.9 


41.1 


7 


a. Arietis 


2 2 


31.0 


30.1 


30.8 


31.7 


32.7 


33.6 


34.3 


34.6 


34.7 


34.5 


8 


9 Eridani 


255 


8.8 


6.8 


7.2 


7.9 


9.0 


10.0 


10.8 


11.3 


11.4 


11.0 


9 


a. Persei 


3 18 


25.9 


23.9 


24.4 


25.5 


26.8 


28.2 


29.3 


30.2 


30.6 


30.5 


10 


a Tauri 


431 


11.7 


10.3 


10.5 


11.0 


11.9 


12.8 


13.7 


14.5 


15.0 


15.2 


11 


|3 Orionis 


5 10 


35.1 


33.7 


33.7 


34.2 


34.7 


35.6 


36.5 


37.3 


37.8 


38.1 


12 


a- Aurigse 


5 10 


36.5 


34.5 


34.6 


35.2 


36.2 


37.5 


38.7 


39.9 


40.7 


41.1 


13 


y Orionis 


5 20 


43.1 


41.7 


41.7 


42.1 


42.8 


43.7 


44.6 


45.4 


46.0 


46.4 


14 


f Orionis 


532 


2.4 


1.0 


1.0 


1.3 


2.0 


2.8 


3.7 


4.5 


5.2 


5.5 


15 


a Orionis 


550 


43.1 


41.8 


41.7 


42.0 


42.7 


43.5 


44.4 


45.3 


46.0 


46.4 


16 


a Argus 


622 


9.2 


6.1 


5.5 


5.4 


6.0 


6.9 


8.1 


9.3 


10.2 


10.6 


17 


a Can. Maj. 


641 


31.6 


30.2 


30.0 


30.1 


30.6 


31.3 


32.2 


33.1 


33.8 


34.3 


18 


eCan. Maj. 


655 


24.1 


22.6 


22.2 


22.2 


22.6 


23.3 


24.2 


25.2 


26.0 


26.5 


19 


a Can. Min. 


734 


59.7 


59.0 


58.7 


58.8 


59.1 


59.8 


60.5 


61.5 


62.3 


63.0 


20 


ft Gemin. 


740 


17.1 


16.3 


16.0 


16.0 


16.4 


17.1 


18.0 


19.0 


20.0 


20.8 


21 


< Argus 


820 


51.4 


49.0 


48.0 


47.3 


47.2 


47.8 


48.9 


50.4 


51.8 


52.8 


22 


* Argus 


9 4 


58.6 


57.9 


57.3 


56.9 


56.8 


57.1 


57.8 


58.9 


60.1 


61.0 


23 


ft Argus 


9 12 


20.6 


18.1 


16.4 


15.1 


14.5 


14.8 


16.0 


17.9 


20.0 


21.7 


24 


a Hydrse 


9 23 


32.5 


32.6 


32.2 


32.0 


32.0 


32.3 


32.9 


33.7 


34.7 


35.6 


25 


a Leonis 


10 3 


59.2 


59.7 


59.3 


59.1 


59.0 


59.2 


59.7 


60.5 


61.4 


62.4 



Had it been thus necessary to get the declination without 
using the chronometer, we should have proceeded as follows : 

Apparent solar time of noon (p. 75) 0* O m (1) 

Approximate longitude = 74 0' W. = (at 15 to 

the hour) 4 56 W. (2) 

Adding (1) and (2) (p. 81) gives approximate 

Greenwich apparent time 4 56 (3) 

Approx, eq. of time, Dec. 17, at 4* 56 W (p. 76) + 4 (4) 
Subtracting l (4) from (3) gives approximate 

G. M. T 4 52 (5) 

Declination interpolated for G. M. T. in line (5) is - 23 22' (6) 

1 The equation is additive to G. M. T., according to the note at 
the foot of p. 76, and therefore to be subtracted from Greenwich 
apparent time. 



92 



NAVIGATION 



APPARENT PLACES OF STARS, 1917 

From Nautical Almanac, p. 95 
FOB THE UPPER TRANSIT AT GREENWICH 





DECLINATION 






No. 






. 


,H 


^ 


rt 




rt 


M 


IN 

CO 


SPECIAL NAME 


MAO. 1 






a 
a 
<-> 


1 


3 
% 


ft 

<J 


i 

% 





1 


o 

Q 


* 






1 




+ 28 


38.2 


38.1 


38.0 


38.0 


38.0 


38.4 


38.5 


38.5 


/ 

38.5 


Alpheratz 


2.2 


2 


+ 58 


41.9 


41.8 


41.7 


41.6 


41.5 


42.0 


42.1 


42.2 


42.2 


Caph 


2.4 


3 


- 18 


26.5 


26.5 


26.5 


26.4 


26.3 


26.0 


26.1 


26.2 


26.2 


Deneb Kaitos 


2.2 


4 


+ 59 


48.7 


48.7 


48.6 


48.4 


48.3 


48.6 


48.8 


48.9 


49.0 


Ruchbah 


2.8 


5 


+ 88 


52.2 


52.2 


52.1 


52.0 


51.8 


52.0 


52.2 


52.4 


52.5 


Polaris 


2.1 


6 


-57 


39.7 


39.7 


39.6 


39.4 


39.2 


39.0 


39.2 


39.3 


39.4 


Achernar 


0.6 


7 


+ 23 


4.5 


4.4 


4.4 


4.3 


4.3 


4.6 


4.7 


4.7 


4.7 


Hamal 


2.2 


8 


-40 


38.3 


38.3 


38.3 


38.2 


38.1 


37.7 


37.8 


38.0 


38.1 


Acamar 


3.0 


9 


+ 49 


34.3 


34.3 


34.3 


34.2 


34.1 


34.3 


34.3 


34.4 


34.5 




1.9 


10 


+ 16 


20.7 


20.7 


20.7 


20.7 


20.7 


20.8 


20.8 


20.8 


20.8 


Aldebaran 


1.1 


11 


- 8 


17.8 


17.8 


17.9 


17.9 


17.8 


17.5 


17.6 


17.7 


17.7 


Rigel 


0.3 


12 


+ 45 


55.0 


55.1 


55.1 


55.1 


55.0 


54.9 


54.9 


55.0 


55.1 


Capella 


0.2 


13 


+ 6 


16.6 


16.5 


16.5 


16.5 


16.5 


16.7 


16.7 


16.6 


16.6 


Bellatrix 


1.7 


14 


- 1 


15.2 


15.3 


15.3 


15.3 


15.3 


15.0 


15.1 


15.1 


15.2 


Alnitam 


1.8 


15 


+ 7 


23.6 


23.5 


23.5 


23.5 


23.5 


23.7 


23.7 


23.6 


23.6 


Betelgeux 


1.0-1.4 


16 


-52 


39.0 


39.2 


39.3 


39.3 


39.2 


38.7 


38.7 


38.9 


39.1 


Canopus 


-0.9 


17 


- 16 


36.1 


36.2 


36.3 


36.3 


36.3 


35.9 


36.0 


36.1 


36.2 


Sirius 


- 1.6 


18 


-28 


51.5 


51.7 


51.7 


51.8 


51.7 


51.3 


51.4 


51.5 


51.6 


Adhara 


1.6 


19 


+ 5 


26.3 


26.2 


26.2 


26.2 


26.2 


26.3 


26.2 


26.2 


26.1 


Procyon 


0.5 


20 


+ 28 


13.6 


13.6 


13.6 


13.7 


13.7 


13.5 


13.5 


13.4 


13.4 


Pollux 


1.2 


21 


-59 


14.4 


14.6 


14.8 


14.9 


14.9 


14.4 


14.4 


14.5 


14.7 




1.7 


22 


- 43 


5.7 


5.9 


6.1 


6.2 


6.2 


5.8 


5.8 


5.9 


6.0 




2.2 


23 


- 69 


22.4 


22.6 


22.8 


22.9 


23.0 


22.5 


22.4 


22.5 


22.7 


Miaplacidus 


1.8 


24 


- 8 


17.9 


18.1 


18.1 


18.2 


18.2 


18.0 


18.0 


18.1 


18.2 


Alphard 


2.2 


25 


+ 12 


22.2 


22.2 


22.2 


22.2 


22.2 


22.2 


22.1 


22.0 


21.9 


Regulus 


1.3 



1 When tlie number in this column is very small, and especially when it is minus, 
the star is very bright. 

It is further to be noted that as we can thus obtain the 
approximate G. M. T., we really know in advance the approx- 
imate moment when the observation should be made. So 
it is unnecessary to get the sextant ready a long time before 
the observation ; and it is, in fact, better to observe at the 
proper predetermined approximate moment rather than to 
wait for the maximum altitude (p. 86). 

When the ship's position at noon can be predicted with fair 
approximation, it is thus possible to have the declination and 
other numbers for calculating the noon-sight also all ready 



OLDER NAVIGATION METHODS 93 

in advance, so that the latitude will be immediately available 
when the noon altitude has been read from the sextant. 

We shall now consider the following example : Off St. 
Paul de Loando, West Africa, approximate latitude 8 55' 
south, approximate longitude 12 55' east, both predicted 
in advance by D. R. for noon on Monday, Dec. 31. The 
altitude of the sun's lower limb is to be measured. Index 
correction is 5'. Height of eye, 26 ft. 
To prepare for the observation, we have, as before : 

Apparent solar time of noon 0* O m (1) 

Approximate D. R. longitude = 12 55' east = (at 15 to 

the hour) 52 E. (2) 

Subtracting (2) from (1) gives approximate Greenwich 

apparent time, Dec. 30 23 8 (3) 

Approximate equation of time, Dec. 30, at 23* 8 W 

(p. 76) - 3 (4) 

Subtracting (4) from (3), having regard to sign of 

(4), gives approximate G. M. T 23 11 (5) 

The navigator will then make the observation when the 
G. M. T. is 23 A 11 TO , as indicated by the chronometer, duly 
corrected for error and -rate. This would of course also be 
noon, or the time when the sun attained its maximum altitude 
for the day. 

Now the dials of chronometers are always divided into 
12 hours, like ordinary watches, although navigators count 
time through 24 hours, as we have seen (p. 75). The 
reason is that the dial would be overloaded with numbers 
if there were 24 hour divisions. Therefore, when we speak 
of the chronometer indicating 23* 11"*, it must be under- 
stood that the actual chronometer indication, or "chro- 
nometer face," as it is sometimes called, would really be 
II 71 ll m ; only, the navigator would call it 23* ll m , astronomic 
time. In this manner civil time still forces its way into 
navigation, by way of the chronometer face. 

To make the observation at the prearranged G. M. T. by 
chronometer it is not desirable to carry that instrument out 
into the sunlight, where the observer stands. It is much 



94 NAVIGATION 

better for the navigator to use his watch, and to calculate in 
advance the "watch time" of the observation. To do this, 
it is merely necessary to compare the watch with the chro- 
nometer, and thus ascertain how much the watch is slow or 
fast of the chronometer. This amount is called "chro- 
nometer minus watch" (abbreviated C. W.) ; and when the 
watch is fast of the chronometer, C. W. is marked with the 
minus sign. 

To obtain the watch time for the observation, we subtract 
C. W. from the G. M. T. In the present case we will 
suppose the watch was 47 m fast of the chronometer. Then 
C. W. = 47 m . To get the watch time for the observa- 
tion we must subtract 47 m from 23 A ll m . Subtracting a 
minus number is equivalent to addition ; and so the watch 
time is 23* ll m + 47 m = 23* 58 TO . The observation would 
be made as nearly as possible 2 m before noon, by the watch. 

In this connection it also becomes of interest to inquire 
how the navigator's watch happened to be 47 m fast of the 
chronometer. It is customary aboard ship to set the deck 
and cabin clocks, and all watches, to the ship's local apparent 
time once a day at least. To do this, we proceed as follows : 

Take from chronometer the G. M. T., corrected for error and rate (1) 
Apply to this G. M. T. the eq. of time, giving Green'h app. time (2) 
Apply to (2) the approximate D. R. longitude, adding it if longi- 
tude is E., which gives ship's apparent time (3) 

And set the watch to the time (3). 

An example of this proceeding can be had from the data on 
p. 93. Suppose the watch was to be set; and the chro- 
nometer time was 23* O ro . We should then prepare to set the 
watch in about 5 m , when the 

G. M. T. by chronometer would be 23* 5" (1) 

Chronometer error (corrected for rate) say 2 (2) 

Corrected G. M. T. by chronometer, (1) +(2) 23 3 (3) 

Equation of time (p. 93) 3 (4) 

Greenwich apparent time, (3) + (4) 23 (5) 

Approximate longitude (p. 93) 52 E. (6) 

Ship's apparent time, (5) + (6) 23 52 (7) 



OLDER NAVIGATION METHODS 95 

And the watch would be set to 23* 52 m , when the chro- 
nometer face was 23 A 5 m ; or, which is the same thing, the 
watch would be set at 8 TO to 12 when the chronometer in- 
dicated 5 minutes past 11. 

Sometimes the navigator wishes the watch to be correct 
by ship's apparent time at noon, but desires to set it right 
half an hour sooner, so as to be free at noon to make an 
observation. In that case he calculates by D. R. what the 
longitude will be at noon, and proceeds practically in the 
same way as before. 

Resuming now the example of p. 93, we are still 
off St. Paul de Loando, and at 2 W before noon by the 
watch (p. 94) the altitude of the sun's lower limb was 
measured. 

Suppose it was found to be 75 34' (1) 

The index correction was 5 (2) 

Adding (1) and (2), with regard to sign of (2), gives 

corrected altitude 75 29 (3) 

Correction from Table 6 +16 (4) 

Correction from Table 7, for 26 ft. height of eye 5 (5) 

Adding (3), (4), (5) gives corrected altitude 75 40 (6) 

Formula (2), p. 89, is the proper one, and the inter- 
polated declination, disregarding sign, is 23 8 (7) 

Latitude, by formula, is (6) + (7) - 90, or 8 48 (8) 

The latitude of the ship is therefore 8 48' south, from the 
above noon-sight observation. The difference of 7' from 
the approximate latitude (p. 93) might easily be caused by 
ocean currents. 

Our next example is a star observation. Position of ship 
by D. R. March 23, 1917, at 6* 3(T ship's time is : latitude 
40 25' N., longitude 46 52' W., so that she is near the turning 
point in the southern "lane route" followed by steamships 
bound from New York to Fastnet in summer. The upper 
transit (p. 89) of Sirius was observed; and the sextant 
altitude was 33 7'. Index correction, 7' ; height of eye, 
24ft. 



96 NAVIGATION 

The calculation is as follows : 

Observed altitude of Sirius 33 7' (1) 

Index correction 7 (2) 

Adding (1) and (2), having regard to minus sign of (2), 

gives corrected altitude 33 (3) 

Correction Tables 6 and 7, combined 6 (4) 

Adding (3) and (4) gives finally corrected altitude .... 32 54 (5) 
Use formula (4), p. 89, because latitude is + and decli- 
nation of Sirius -. We have 90 (6) 

Subtract (5) from (6), giving (90 - altitude) 57 6 (7) 

Declination of Sirius (p. 92), disregarding sign, is. . . 16 36 (8) 
Subtract (8) from (7), giving (90 altitude declina- 
tion), or the latitude 40 30 (9) 

Ship's latitude at the moment of observation was therefore 
40 30' N. 

In making such a star observation, it is of course possible 
to follow the star with the sextant until it begins to 
dip (p. 86) toward the horizon exactly as we have ex- 
plained for the sun. But it is preferable to prepare for the 
observation in advance, and to make it at a definite prede- 
termined minute by the navigator's watch. To make such 
preparation, it is necessary to use pages 96 and 97 of the 
Nautical Almanac, parts of which pages are reprinted here 
(pp. 97, 98). 

The almanac page 96 gives for all the bright stars the 
G. M. T. of upper transit (p. 89) at Greenwich, for the first 
day of each month. And it will be noticed that the upper 
transit is here called "meridian transit," which is practically 
another name for the same thing. Almanac page 97 (our 
p. 98) then gives a subtractive correction, applicable to the 
numbers on page 96, to make them correct on days of the 
month other than the 1 st . 

Another small correction is still required to make the 
numbers right in the approximate D. R. longitude of the ship, 
instead of the longitude of Greenwich, as used on almanac 
page 96. This correction is subtractive, if the ship is in west 
longitude, and additive, if she is in east longitude ; and the 



OLDER NAVIGATION METHODS 



97 



MERIDIAN TRANSIT OF STARS, 1917 

From Nautical Almanac, p. 96 
GREENWICH MEAN TIME OF TRANSIT AT GREENWICH 



CONSTELLA- 
TION 

NAME 


MAO. 




















z 

3 


t 

h 


3 
S3 


g 

< 


> 

<! 
S 


fc 

H 

CO 


g 
O 


o 

Z 


I 

Q 






h m 


h m 


h m 


h m 


h m 


h in 


h m 


h m 


h m 


a Androm. 


2.2 


5 21 


3 19 


1 29 


23 23 


21 25 


13 22 


11 24 


9 22 


7 24 


ft Cassiop. 


2.4 


5 22 


3 20 


1 30 


23 24 


21 26 


13 22 


11 24 


9 22 


7 24 


PCeti 


2.2 


5 56 


3 54 


2 4 


f o a! 

(23 SSS 


22 


13 57 


11 59 


9 57 


7 59 


S Cassiop. 


2.8 


6 37 


4 35 


2 45 


43 


22 41 


14 38 


12 40 


10 38 


8 40 


a Urs. Min. 


2.1 


6 47 


4 45 


2 54 


52 


22 50 


14 49 


12 51 


10 49 


8 51 


a Eridani 


0.6 


6 51 


4 49 


2 59 


57 


22 55 


14 52 


12 54 


10 52 


8 54 


a Arietis 


2.2 


7 19 


5 17 


3 27 


1 25 


23 23 


15 20 


13 22 


11 20 


9 22 


6 Eridani 


3.0 


8 12 


6 10 


4 20 


2 18 


20 


16 12 


14 14 


12 12 


10 14 


a Persei 


1.9 


8 35 


6 33 


4 43 


2 41 


43 


16 35 


14 38 


12 36 


10 38 


a Tauri 


1.1 


9 47 


7 46 


5 55 


3 54 


1 56 


17 48 


15 50 


13 48 


11 50 


ft Orionis 


0.3 


10 27 


8 25 


6 35 


4 33 


2 35 


18 27 


16 29 


14 28 


12 30 


a Aurigse 


0.2 


10 27 


8 25 


6 35 


4 33 


2 35 


18 27 


16 29 


14 28 


12 30 


y Orionis 


1.7 


10 37 


8 35 


6 45 


4 43 


2 45 


18 37 


16 39 


14 38 


12 40 


e Orionis 


1.8 


10 48 


8 46 


6 56 


4 54 


2 56 


18 49 


16 51 


14 49 


12 51 


a Orionis 


1.0-1.4 


11 7 


9 5 


7 15 


5 13 


3 15 


19 7 


17 9 


15 7 


13 9 


a Argus 


-0.9 


11 38 


9 36 


7 46 


5 44 


3 46 


19 39 


17 41 


15 39 


13 41 


a Can. Maj. 


- 1.6 


11 57 


9 55 


8 5 


6 3 


4 5 


19 58 


18 


15 58 


14 


e Can. Maj. 


1.6 


12 11 


10 9 


8 19 


6 17 


4 19 


20 12 


18 14 


16 12 


14 14 


o Can. Min. 


0.5 


12 51 


10 49 


8 59 


6 57 


4 59 


20 51 


18 53 


16 52 


14 54 


ft Gemin. 


1.2 


12 56 


10 54 


9 4 


7 2 


5 4 


20 57 


18 59 


16 57 


14 59 


e Argus 


1.7 


13 36 


11 34 


9 44 


7 42 


5 44 


21 37 


19 39 


17 37 


15 39 


A Argus 


2.2 


14 20 


12 19 


10 28 


8 27 


6 28 


22 21 


20 23 


18 21 


16 23 


ft Argus 


1.8 


14 28 


12 26 


10 36 


8 34 


6 36 


22 28 


20 30 


18 28 


16 31 


a Hydras 


2.2 


14 39 


12 37 


10 47 


8 45 


6 47 


22 40 


20 42 


18 40 


16 42 


a Leonis 


1.3 


15 19 


13 17 


11 27 


9 25 


7 27 


23 20 


21 22 


19 20 


17 22 



amount of it is 10* for every 15 in the ship's longitude. 
After it has been applied, the result will be the ship's mean 
solar time of the star's upper transit. 

As an example, let us take the preparation for the fore- 
going observation of Sirius, or a Can. Maj. We have : 
G. M. T. of upper transit, March 1, from almanac 

page 96 above 8* 5"* (1) 

Correction for 23d day of month, from almanac 

page 97 (our p. 98) - 1 27 (2) 

Correcting (1) with (2), having regard to - sign of (2) 6 38 (3) 
Further correction for longitude 46 52' W., at 10* per 

15 of longitude, approximately , 1 (4) 

Subtracting (4) from (3) gives ship's mean solar time 

of the observation 6 37 (5) 



98 



NAVIGATION 



MERIDIAN TRANSIT OF STARS, 1917 
From Nautical Almanac, p. 97 

CORRECTIONS TO BE APPLIED TO THE MEAN TIME OF TRANSIT ON 
THE FIRST DAY OF THE MONTH, TO FIND THE MEAN TIME OF 
TRANSIT ON ANY OTHER DAY OF THE MONTH 



DAY OF 

MONTH 


CORRECTION 


DAY OF 
MONTH 


CORRECTION 


DAY OF 
MONTH 


CORRECTION 




h m 




h m 




h m 


1 


-0 


11 


-0 39 


21 


-1 19 


2 


4 


12 


43 


22 


1 23 


3 


8 


13 


47 


23 


1 27 


4 


12 


14 


51 


24 


1 30 


5 


16 


15 


55 


25 


1 34 


6 


-0 20 


16 


-0 59 


26 


-1 38 


7 


24 


17 


1 3 


27 


1 42 


8 


28 


18 


1 7 


28 


1 46 


9 


31 


19 


1 11 


29 


1 50 


10 


35 


20 


1 15 


30 


1 54 


11 


-0 39 


21 


- 1 19 


31 


- 1 58 



NOTE. If the quantity taken from this Table is greater than the 
mean time of transit on the first of the month, increase that time 
by 23" 56 m and then apply the correction taken from this Table. 



The actual observation was made at 6 A 30 m , ship's time, 
as indicated by the navigator's watch. The difference of 
7 m between 6* 30", and 6* 37 m in line (5) above, is due to the 
equation of time (p. 77), which is 7 on March 23. This 
7 m , if applied (with its proper sign from the almanac) to 
line (5) above, will give the ship's apparent time; and we 
have seen that watches and clocks on board are usually 
kept set to apparent and not mean ship's time (p. 94). 

To complete this part of our subject, we have still to con- 
sider a few additional points of interest. For instance, a 
star chosen for observation may be one of the planets : 
Mars, Jupiter, or Saturn. These look like very bright stars 
in the sextant telescope; and calculations depending on 
them are similar to those described for stars. The planetary 
declinations and the G. M. T.'s of their upper transits are 
given in the almanac, but not on the pages reprinted here. 



OLDER NAVIGATION METHODS 99 

The moon is now so rarely observed that we have not given 
examples of lunar observations. 

Sometimes an "ex-meridian" observation of the sun or 
a star is made at a time very near the upper transit, on a 
day when the actual transit observation could not be secured 
because of clouds. There are special tables 1 for calculating 
observations of this kind; but we have not included them 
here because all such observations can be satisfactorily 
treated by a new general method to be explained later 
(p. 108). 

Having now fully treated the older standard method of 
determining the ship's latitude, let us next consider the older 
way of obtaining the longitude. This cannot be done when 
the sun (or a star) is near its maximum altitude, as already 
explained (p. 88). The most favorable opportunity occurs 
when the observed object bears (p. 44) east or west; but 
it is not always possible to get the observation on such a 
bearing. In that case, the longitude observation, often 
called a "time-sight," must be taken when the sun is near 
the desired bearing, but always avoiding, if possible, observa- 
tions at very low altitudes. And if a very low altitude has 
been observed in an emergency, it can sometimes be checked 
by a later observation at a better altitude. 

The principle on which the time-sight depends is simple. 
Calculations based on the measured altitude make known 
the ship's mean time at the moment of observation. At 
the same moment the chronometer face (p. 93), duly cor- 
rected for error and rate, tells us the G. M. T. The 
difference between the two times then gives us the longitude 
(see p. 82). 

The calculations for this problem are made by means of 
Table 4 (trigonometric logarithms) and Table 10 ("haver- 
sines"). These haversines (abbreviated hav.) are really 
additional trigonometric logarithms; and Table 10 gives 
in every case not only the haversine itself, which is really 
1 Tables 26 and 27 of Bowditch's "Navigator," for instance. 



100 NAVIGATION 

a logarithm, but also, in the adjoining heavy type col- 
umns, the number (abbreviated No.) of which the haver- 
sine is the log. This additional heavy type number is not 
given throughout the entire table, but only when necessary 
for working Sumner line calculations (see Chapter IX, 
p. 108). It is not needed in working time-sights. 

The argument (p. 10) of the haversine table is a double 
argument, not to be confounded with the pairs of arguments 
already explained (p. 11). In the haversine table, the argu- 
ment is generally given in degrees and minutes, as well as 
(for convenience) in hours and minutes of time, allowing 
the usual 15 to each hour, etc. 

We shall now solve our time-sight problem for the sun; 
and in doing so shall make use of two angles not hitherto 
employed: the "polar distance" (abbreviated p), and the 
"half sum" (abbreviated s). We shall also, for brevity, 
indicate the ship's apparent solar time by T. Then we 
have the following formulas : 

If lat. and dec. are both + or both . . p = 90 dec. (1) 

If lat. and dec. are one + and one . . . p = 90 + dee. (2) 

In every case s = % (alt. + lat. + p) (3) 

If time-sight was made before noon, ship's time, 

hav. (24* T) = sec lat. + esc p + cos s + sin (s alt.) (4) 
If time-sight was made after noon, ship's time, 

hav. T= sec lat. + esc p + cos s + sin (s alt.) (5) 

In using these formulas, we have to choose between (1) 
and (2), and also between (4) and (5). Formula (3) is 
always used. No attention need be given to the signs 
of the declination or latitude except in choosing between 
formulas (1) and (2) for calculating p; and in choosing 
between (4) and (5), we have merely to note whether the 
time-sight was taken in the forenoon or afternoon by ship's 
time. 

We also desire to emphasize especially that these formulas 
presuppose the latitude to be known. This is merely 
another application of the principle (p. 88) that both lati- 



OLDER NAVIGATION METHODS 101 

tude and longitude cannot be determined from a single 
observation. It follows that in using this method we must 
first determine the latitude by a noon-sight before we can 
calculate the time-sight for longitude. If the time-sight 
was taken in the afternoon, the noon-sight will naturally 
have preceded it, and the ship's latitude at noon will be 
known. This noon latitude must then be carried forward 
to the moment of the afternoon time-sight by D. R. methods 
(p. 7) ; and the latitude thus obtained must be used for 
calculating the time-sight. 

But if the time-sight was a forenoon observation, it cannot 
be properly calculated until noon, when the latitude will 
be determined. After that, the latitude can be carried 
backwards by D. R. to the moment of the forenoon time- 
sight, and the latter can be calculated. 

But if the navigator, because of emergency, needs his 
longitude at once, after taking the forenoon time-sight, he 
must obtain the latitude by a D. R. calculation based on the 
last good noon-sight. Most navigators calculate morning 
time-sights in this way, and then repeat the calculation 
after the new noon-sight has been obtained. The latter 
calculation will be preferable to the former, because the 
further the latitude is carried along by D. R., the less accurate 
will it be. And any error in the latitude used in the calcula- 
tion will impress a consequent error on the calculated longi- 
tude. 

We shall now work some time-sight examples. On board 
ship, at sea, Dec. 18, 1917, in the afternoon, D. R. latitude 
42 20' N., D. R. longitude 35 16' W., the altitude of sun's 
lower limb was observed to be 14 19'. The time was taken 
with the navigator's watch, and was 2 h 29 m 58*. A com- 
parison of the watch and ship's chronometer gave C. W. = 
2 h 27 m 8*. The chronometer correction was 2 m 8* slow of 
G. M. T. The index correction of the sextant was + 4' ; 
height of eye, 24 ft. Calculate the ship's longitude. 

We have first to find, for the moment of the observation. 



102 NAVIGATION 

values of the declination and equation of time. To do this, 
we have : 

Watch time of observation 2 29" 58 (1) 

C. -W 2 27 8 (2) 

Adding (1) and (2) gives chronometer time of 

observation 4 57 6 (3) 

Chronometer correction, slow 2 8 (4) 

Adding (3) and (4) gives G. M. T. of observation 4 59 14 (5) 

For the G. M. T. (5) we interpolate the declina- 
tion (p. 76), finding - 23 24' (6) 

and for the same G. M. T. we interpolate the 

equation of time + 3 W 21* (7) 

Now, adding (5) and (7) gives Greenwich ap- 
parent time of observation 5* 2 m 35 (8) 

Next we inspect the formulas (p. 100), choosing (2) be- 
cause latitude is + and declination , and (5) because the 
sight was an afternoon one. 

We now have, from line (6), declination (disregard- 
ing sign) 23 24' (9) 

to which, by formula (2), we add , 90 (10) 

giving p 113 24 (11) 

The observed altitude was 14 19 (12) 

Index correction +4 (13) 

Adding (12) and (13) gives corrected altitude 14 23 (14) 

Correction, Table 6 +12 (15) 

Correction, Table 7 - 5 (16) 

Adding (14), (15), (16) gives finally corrected altitude 14 30 (17) 

The latitude by D. R. is 42 20 (18) 

Adding (11), (17), (18) gives ' 170 14 (19) 

Halving (19) gives (by formula (3), p. 100) s 85 7 (20) 

Subtracting (17) from (20) gives (s - alt.) 70 37 (21) 

Next we apply formula (5), p. 100. We have: 

sec lat. (18) from Table 4, page 238 0.13121 (22) 

esc p (11) from Table 4, page 219 0.03727 (23) 

cos s (20) from Table 4, page 200 8.93007 (24) 

sin (s - alt.) (21) from Table 4, page 215 9.97466 (25) 

sum (22) to (25) = hav. T, by formula (5) 9.07321 ' (26) 

1 This sum has been diminished by 10 arbitrarily (see p. 25), 
which must always be done when the sum of logs is larger than 10. 



OLDER NAVIGATION. METHODS 103 

TV corresponding to (26) from Table 10, page 260, is 2* 40 W 59* (27) 
Greenwich apparent time (8) by watch and 

chronometer is 5 2 35 (28) 

Subtract (27) from (28), giving time difference 

between ship and Greenwich 2 21 36 (29) 

Turning (29) into degrees with Table 9, page 249, 

gives 35 24' W. (30) 

and (30) is the ship's longitude from this time-sight. 

Upon comparing the D. R. longitude (35 16' W.) with the 
result of the time-sight (35 24' W.), we find that the ship 
is 8' west of her D. R. position. This means, of course, that 
there has been a westerly "set" of current in the interval 
between the last accurate determination of longitude and 
the present one. It would be proper for the navigator to 
calculate from this the amount of westerly drift per hour, 
and to allow for it in carrying forward his longitude by D. R. 
from the present time-sight. It is also clear that the 
northerly or southerly set of the current can be similarly 
measured and allowed for by comparing the D. R. latitude 
with the latitude from a noon-sight (cf. p. 95). It is the 
general custom of navigators to ascribe such differences to 
ocean currents, never to uncertainty in the astronomic results. 
Dead reckoning is never allowed any weight as against a 
sextant observation. 

The reader will have noticed that the foregoing calculation 
has been made in great detail, so that a beginner may have 
no difficulty in understanding it. But a practiced navigator 
would of course work the calculation in a much more con- 
densed form, in such a way as to bring the logarithms next 
to the numbers to which they belong. We shall therefore 
now repeat the same example in such a condensed form : 

1 If the observation had been made before noon, we should have 
used formula (4) and should here have obtained 24* T, instead 
of T. This 24* - T would then be subtracted from 24*, to get 
T, before continuing the calculation. Thus the form of calculation 
would contain another line between (27) and (28), in the case of 
a forenoon observation. 



104 



NAVIGATION 



TIME-SIGHT, CONDENSED FORM. SUN 



Watch time : 
C. - W. : 

Chr. time : 
Chr. corr'n : 



2 29 58' (1) 



2 27 
4 57 

+ 2 



6 
8 
14 
21 
35 



G. M. T. : 18> 4 59 

Eq. of time : + 3 

G. app. time : 5 2 

Decl. 18 th , 4* : 23 23'.7 
H. D.: 0.1 

Decl. 4 59 : 23 24 

p: 113 24 



(2) 
(3) 
(4) 
(5) 
(7) 
(8) 



Obs'd alt. : 


14 19' (12) 


Index : 


+ 4 (13) 


Table 6 : 


+ 12 (15) 


Table 7 : 


- 5 (16) 


Corr'd alt. : 


14 30 (17) 



(6) 
(11) 



Eq. time, 18 th , 4* : + 3 m 22*.3 
H. D.: 1.2 

Eq. time, 4 s 59 : +3 21.1 



(7) 



Corr'd alt. 
Lat., D. R. 
p: 
sum of 3 : 
s : 
s alt. : 

By chron., 


r 14 30' (17) 
: 42 20 (18) sec lat. : 
113 24 (11) esc p: 


0.13121 (22) 
0.03727 (23) 

8.93007 (24) 
: 9.97466 (25) 


2)170 14 (19) 
85 7 (20) cos s : 
70 37 (21) sin (s-alt.): 
sum of 4 : 

T = ship's app. time : 
Greenwich app. time : 
Longitude : 
or: 


9.07321 (26) = hav. T 
(or 24* - T) * 
2 40" 59* (27) 
5 2 35 (8) 


2 ft 21" 36* (29) 
35 24' W. (30) 



When the object observed is a star or planet, the choice 
between formulas (4) and (5), p. 100, is not quite the same 
as in the case of a solar time-sight. We must use (4) if there 
is any east in the star's bearing at the moment of observation ; 
and (5), if there is west in the bearing. The more nearly the 
star bears due east or west, the more accurate will be the 
resulting longitude. The use of formulas (1), (2), and (3) 
is the same as for the sun ; but T, in the case of a star, is no 
longer the ship's apparent solar time. Instead, it is called 



1 See p. 103, footnote. 



OLDER NAVIGATION METHODS 105 

the star's "hour-angle." To get the longitude, we must 
first (p. 85) calculate the Greenwich sidereal time corre- 
sponding to the G. M. T. of the observation, as taken from 
the chronometer, duly corrected for error and rate; and 
then use the following formulas : 

(6) Greenwich sid. time 1 right-ascension of star = Greenwich 
hour-angle. 

, | West long. = Greenwich hour-angle - T, 
\ East long. = T Greenwich hour-angle. 

As an example of a star observation we shall take the 
following : 

At sea, just before sunrise, Dec. 17, 1917, off Cape Agulhas, 
latitude by D. R. 35 20' S., longitude by D. R. 20 41' E., 
the altitude of Sirius was measured, and found to be 40 3'. 
The star bore west, and the height of eye was 22 ft. Index 
correction was -f 5'. Time by watch, 16* 29" 1 48*, or 4* 29"* 
48' A.M., civil time, Dec. 18; C. - W., - l h 2Z m 50'; chro- 
nometer fast of G. M. T. 2 m 28*. 

The calculation would proceed thus: 

Watch time of observation 16* 29" 48* (1) 

C. - W - 1 23 50 (2) 

Adding (1) and (2), having regard to sign of (2), 

gives chronometer time of observation 15 5 58 (3) 

Chronometer correction, fast 2 28 (4) 

Adding (3) and (4), having regard to - sign of (4), 

gives G. M. T. of observation 15 3 30 (5) 

Right ascension mean sun, Greenwich mean noon, 

Dec. 17 (p. 83) 17 42 10 (6) 

Correction for " time past noon " (see p. 84) .... 2 28 (7) 

Adding (6) and (7) gives right ascension of mean 

sun 17 44 38 (8) 

Adding (5) and (8) (see p. 85) gives Greenwich 

sidereal time of the observation 8 l 48 8 (9) 

Right ascension of Sirius, Dec. 17, is (p. 91) .... 6 41 34 (10) 

Subtracting (10) from (9) gives Greenwich hour- 
angle (formula (6), above) 2 6 34 (11) 

1 24 A may always be added or dropped here, if necessary. 



106 NAVIGATION 

Next we calculate T by formula (5), p. 100. We have: 

Declination of Sinus, Dec. 17 (p. 92) - 16 36' (12) 

By formula (1), p. 100, subtract (12) from 90, 

without attention to sign of (12), giving p. . 73 24 (13) 

The observed altitude was 40 3 (14) 

The index correction was +5 (15) 

Table 6 correction - 1 (16) 

Table 7 correction 5 (17) 

Adding (14), (15), (16), (17), having regard to 

signs, gives corrected altitude 40 2 (18) 

The latitude by D. R. was 35 20 (19) 

Adding (13), (18), and (19) gives 148 46 (20) 

Halving (20) gives s. 74 23 (21) 

Subtracting (18) from (21) gives (s - altitude) . . 34 21 (22) 

Now applying formula (5), page 100, we have : 

sec latitude (19) from Table 4, page 231 0.08842 (23) 

esc p (13) from Table 4, page 212 0.01849 (24) 

cos s (21) from Table 4, page 211 9.43008 (25) 

sin (s - altitude) (22) from Table 4, page 230 9.75147 (26) 

Summing (23) to (26) gives hav. T, by form. (5) . . 9.28846 1 (27) 
7 12 corresponding to (27), from Tab. 10, p. 263 is . . 3* 29 m 14* (28) 
Difference between (28) and (11) is the longi- 
tude by formula (7), page 105 1 22 40 E. (29) 

Turning (29) into degrees with Table 9, page 

249, gives 20 40' E. (30) 



The D. R. longitude, 20 41' E., was therefore within 1' of 
the longitude from this time-sight, and this shows that the 
ship has not been affected by ocean currents since the last 
observation. It is also interesting to note how near sunrise 
the observation was made. The twilight must have been 
quite strong, and the star therefore dim. But star observa- 
tions can be made best in twilight because the horizon line 
can then be seen distinctly. 

1 This sum has also been diminished by 10 (see footnote, p. 102). 

2 Might be 24* T, if the star bore E. instead of W. (see footnote, 
p. 103). 



OLDER NAVIGATION METHODS 



107 



The foregoing example can of course also be arranged in 
condensed form, as follows : 

TIME-SIGHT, CONDENSED FORM. STAR 

Watch time : 
C. - W. : 

Chr. time : 

Chr. eorr'n : 

G. M. T. : 

R. A. mean sun : 

Corr'n, past noon : 

Greenw'h sid. time : 

R. A. of Sirius : 

Greenwich hour-ang. 

T., from (27) : 

Long. : 

or: 



R. A. of Sirius : 

Dec. of Sirius : 

p: 

sec lat. : 

esc. p : 

cos s: 

sin (s alt.) : 

sum of 4 : 



16* 29" 48' (1) 


Obs'dalt. : 40 3' 


(14) 


-1 


23 


50 (2) 


Index : 


+ 5 


(15) 


15 


5 


58 (3) 


Table 6 : 


-1 


(16) 




- 2 


28 (4) 


Table 7 : 


-5 


(17) 


15 


3 


30 (5) 


Corr'd alt. : 40 


2 


(18) 


17 


42 


10 (6) 


Lat. D. R. : 35 


20 


(19) 




2 


28 (7) 


p: 73 


24 


(13) 


8 


48 


8 (9) 


sum: 2)148 


46 


(20) 


6 


41 


34 (10) 


s: 74 


23 


(21) 


: 2 


6 


34 (11) 


(s - alt.) : 34 


21 


(22) 


3 


29 


14 (28) 








1 


22 


40 E. (29) 










20 


40' E. (30) 












6* 41 34' 


(10) 










- 16 36' 


(12) 










73 24 


(13) 










0.08842 


(23) 










0.01849 


(24) 










9.43008 


(25) 










9.75147 


(26) 







9.28846 (27) = hav. T (or 24* - T) 



Having now fully explained both the noon-sight and the 
time-sight, we shall close this chapter with a strong recom- 
mendation to young navigators to familiarize themselves with 
the observation of stars. These always furnish a valuable 
check on sun observations : and at times of danger may save 
the ship when clouds have obscured the sun for days, and 
clearing occurs after sunset. It is easy to learn to know the 
principal stars from Jacoby's "Astronomy," Chapter III, 
"How to Know the Stars." 



1 See footnote, p. 103. 



CHAPTER IX 
NEWER NAVIGATION METHODS 

THE reader may have noticed in Chapter VIII that there 
is a very definite difference between the determination of 
latitude by a noon-sight and longitude by a time-sight : for 
the latitude is obtained without previous knowledge of the 
longitude; but to get the longitude, a previous knowledge 
of the latitude is essential. This is, of course, a decided 
disadvantage in determining longitude, nor is there any 
practicable direct way to get the longitude without first 
knowing the latitude. 

We have also seen (p. 101) that any existing uncertainty 
in our knowledge of the latitude will produce an error in the 
longitude computed from a time-sight. In situations of 
danger it is important to ascertain how great this longitude 
error may be. Suppose, for instance, we have calculated 
a tune-sight with a D. R. latitude that we suspect may be 
as much as 10' too small ; and we wish to know how much 
our computed longitude may have been thereby put wrong. 
The obvious way to find out is to recompute the longitude 
with an assumed latitude 10' larger than the D. R. latitude. 
The resulting longitude will then show the extreme range 
of error that must have been produced if the D. R. latitude 
was 10' too small. 

A third calculation, with an assumed latitude 10' smaller 
than the D. R. latitude, will similarly exhibit the extreme 
possible range of longitude error in the other direction. 
Thus these two extra calculations will show the limits of 
longitude error that might be caused by a range of 20' in 
the possible error of the D. R. latitude. 

108 



NEWER NAVIGATION METHODS 109 

This rather obvious procedure was probably used long 
ago by more than one intelligent navigator ; but it was first 
published in 1837 by Thomas H. Sumner, an American 
merchant captain. He used the method in dramatic cir- 
cumstances of great danger ; and he brought his ship safely 
into port. According to his own account, he made three 
calculations of the longitude, using three assumed latitudes 
differing by 10', and he of course obtained three different 
longitudes. He then marked or plotted (p. 55) on his chart 
the point indicated by the first assumed- latitude and its 
computed longitude. At this point the ship must have been 
located, if the first assumed latitude had been correct. The 
other two latitudes, with their computed longitudes, indicated 
two more points on the chart ; and at one of these points the 
ship must have been, if either of these additional latitudes 
was correct. 

Sumner found that the three points on the chart lay in a 
straight line; and it became at once evident that whatever 
latitude he might assume (within reason) he would always 
get a point on the same straight line, after computing the 
longitude. In other words, although he did not know his 
latitude accurately, and so could not compute his longitude 
accurately, yet he had found a straight line on the chart 
upon which his ship was surely situated. 

Such a line can always be found in the way Sumner found 
it, or in some preferable modern way; and such a line we 
shall call a "Sumner line," though some writers on naviga- 
tion prefer to call it a "line of position." 

On the occasion of laying down his line, Sumner found that 
it passed directly through Small's Light, near the Irish coast ; 
and as the line bore E.N.E. on his chart, he simply put 
the ship on that course, and in less than an hour he "made" 
Small's Light, actually bearing E.N.E. E., and, as he says, 
"close aboard." He had had no observations after passing 
longitude 21 W., until the morning of Dec. 17, when these 
historic events occurred. He was off a rocky lee shore, in 



110 NAVIGATION 

the midst of a winter gale, after crossing the Atlantic ; only 
a seaman can understand the relief he must have felt when 
that light suddenly appeared off the bow. 

We have given this account of Sumner's experience to 
impress on the young navigator that he must positively 
familiarize himself with the Sumner method of navigation. 
Should we be so fortunate as to have any experienced navi- 
gator among our readers, we ask him to try the Sumner 
method once more, in the manner explained below, even if 
he may have found it troublesome in the past on account of 
certain difficulties in its application. For the Sumner 
method is the best method of navigation on all oceans and 
at all times : even when a noon-sight is available for latitude, 
it is better to treat it as a Sumner observation, and work 
out the Sumner line. 

The principal objection urged against it by certain prac- 
tical navigators arises from the small scale of existing ocean 
track charts, on which a distance of 10' is represented by 
about - inch. A line like Sumner's, 20' long, would have 
only a length of \ inch on the chart ; and such a little line 
would not be long enough to show accurately the direction 
in which it pointed. When near a coast, as in Sumner's 
case, this difficulty disappears, because navigators always 
have (or always should have and use) the large scale charts 
that can be obtained for coastwise waters. 

But it is inconvenient for navigators to begin using a 
method off the coast, on the last day of a voyage, different 
from the form employed for many days at sea. Therefore, 
some authorities recommend the construction of a special 
large scale chart, with its latitude and longitude lines, each 
tune an observation is made throughout the voyage, so that 
the Sumner line can always be drawn on a sufficiently large 
scale. It is no wonder that navigators have not generally 
adopted this somewhat laborious proceeding; and in the 
method given below we shall utilize the Sumner idea without 
requiring any lines to be drawn on charts. 



NEWER NAVIGATION METHODS 111 

Another objection to Sumner navigation is that it requires 
too much calculation ; three longitude calculations for one 
observation, as Sumner practiced it. This objection is also 
quite removed now by the use of suitable tables such as we 
give in the present volume. 

But before proceeding to explain these tables, we must 
outline briefly the real principle on which rests the com- 
plete utilization of the Sumner method on the open sea. 
There the navigator wants to know the ship's position in 
both latitude and longitude; and will not be satisfied with 
a mere line, with the ship "somewhere on the line." Along 
the coast such a line might help him to find Small's Light ; 
but he is not looking for coast lights at sea. 

And the Sumner method takes care of this matter in the 
simplest possible way. We have seen (p. 88) that two 
different observations are always necessary by any method 
to get both latitude and longitude. But two such observa- 
tions by the Sumner method give two different lines on the 
chart : arid as the ship must be located on both lines, her 
actual position must be at their point of intersection. We 
shall show how the required latitude and longitude of the 
ship at the point of intersection can be found by a simple 
calculation, without the drawing of any lines on the chart. 

Coming now to the modern method of calculating a Sum- 
ner line, we must first state a general fundamental principle 
that may be easily verified by geometrical considerations. 
The true bearing (p. 44) of a Sumner line on a chart is 
always 90 greater than the true bearing or azimuth (p. 44) 
of the sun (or star) at the moment of observation. Or, in 
other words, the Sumner line bears at right angles to the 
sun at the time of observation. 

We shall show how the bearing or azimuth of the sun can 
always be found from suitable "agimuth tables"; but the 
Sumner line is not completely known from its bearing alone. 
To locate it properly it is necessary to know in addition the 
latitude and longitude of some point on the line, which we 



112 NAVIGATION 

will call a "Sumner point." Then, knowing such a point of 
the line, and the bearing of the line, we may say we know the 
line completely, and, if necessary, could draw it on a chart. 

Now to find the required Sumner point. We always have 
the D. R. position of the ship at the moment of observation ; 
which we will call the "D. R. point." It is easy to find 
out if the D. R. point is also a Sumner point. It is merely 
necessary to calculate what the sun's altitude would be for 
a ship at the D. R. point, and then compare this calculated 
altitude with the one actually observed. If the D. R. point 
was really a Sumner point (which will rarely happen), the 
two altitudes will agree ; if not, the amount of disagreement 
will show how far the D. R. point is distant from the nearest 
Sumner point. 1 

The first step, then, in Sumner navigation, is the calcula- 
tion of the altitude, supposing the ship to be at the D. R. 
point at the moment of observation. To do this for a sun 
observation, we first calculate the Greenwich apparent time 
(abbreviated G. A. T.) of the observation, just as was done 
in the case of a time-sight on p. 102. To this G. A. T. we 
then add the ship's D. R. longitude, if east, or subtract it, if 
west, to get T (p. 100), the ship's apparent time of the ob- 
servation. We then use the formulas on p. 113, in which 
X and Z are "auxiliary angles" required in the calculations, 
but not otherwise of special interest. These formulas are 
called the " cosine-haversine " formulas. 

There are several other sets of formulas with which the 
same problem can be solved. One set, called the " haversine " 
formulas, involves the use of haversines only; another, 
called the "sine-cosine" formulas, solves the problem with 
sines and cosines. But neither is preferable to the following 
cosine-haversine set. 

1 This method is often called the Marcq Saint, Hilaire method ; 
but it should probably be credited to Lord Kelvin, who published 
" Tables for Facilitating Sumner's Method at Sea " in 1876. These 
tables follow the method described above. 



NEWER NAVIGATION METHODS 113 

If observation was made before noon, ship's time, 

hav. -X" = cos lat. + cos dec. + hav. (24* - T), (1) 

If observation was made after noon, ship's time, 

hav. X = cos lat. + cos dec. + hav. T, (2) 

lat. dec. = diff. 1 of lat. and dec., if both are + or both , (3) 

lat. dec. = sum 1 of lat. and dec. if one is + and one , (4) 

No. hav. Z = No. hav. (lat. - dec.) + No. hav. X, (5) 

Alt. = 90 - Z. (G) 

Now we can compare the altitude computed by formula 
(6) with the observed altitude, fully corrected for index 
error, etc. The difference between the two altitudes in 
minutes will be the distance in miles of the nearest Sumner 
point from the D. R. point, for the minute and nautical 
mile here correspond, as they do in the case of differences of 
latitude (p. 15). The bearing of the Sumner point from the 
D. R. point will be the same as the sun's azimuth if the ob- 
served altitude is greater than the computed altitude : but if 
the observed altitude is less than the computed, the bearing of 
the Sumner point will be 180 greater than the sun's azimuth. 

The bearing and distance of the Sumner point from the 
D. R. point once known, it is easy, by means of the traverse 
table (p. 10), to obtain the latitude and longitude of the 
Sumner point from the known latitude and longitude of 
the D. R. point ; or, which is the same thing, from the ship's 
D. R. latitude and longitude. 

Before giving examples of these calculations, it remains 
to show how the sun's bearing or azimuth can be taken from 
Table 11 (p. 284), called the azimuth table. The pair of 
arguments (p. 11) for entering this table are: first, in the 
left-hand column, the declination, which is here used without 
regard to its sign; and second, in the four topmost hori- 

1 In using formulas (3) and (4), pay no attention to + or 
signs after the right formula is once chosen. The difference between 
latitude and declination is always taken by subtracting the smaller 
from the larger ; and the sum by adding them, without regarding 
their + or signs. Cf. also p. 89. 



114 NAVIGATION 

zontal lines, T (p. 100), the ship's apparent time at the 
moment of observation. 

Having found this pair of arguments, we look in the 
column under T, and in the horizontal line opposite the 
declination. There we find an "index number." Next we 
look up the altitude, as computed by formula (6), page 113, 
in the right-hand column of the azimuth table, and follow 
along the horizontal line belonging to that altitude, until 
we reach a number equal (or nearly equal) to the index 
number. Then we go down the column containing this 
second appearance of the index number, and find the azi- 
muth at the bottom of the page. The table gives approxi- 
mate azimuths only, but the approximation is sufficient for 
our present purpose. 

The azimuths at the bottom of the page appear in four 
horizontal lines, of which the upper two belong to forenoon 
observations, and the lower two to afternoon observations. 
All azimuths are counted from the north, through east, 
south, and west, from to 360, like compass courses in 
United States Navy practice (p. 41). It is important for 
the navigator to record, at the time of observation, the word 
"forenoon" or "afternoon," and also the sun's roughly 
approximate bearing, to aid in choosing which of the azi- 
muths at the bottom of the tabular page is the right one. 
The record showing whether the observation was made in 
the forenoon or afternoon limits the choice to two of the lines 
of azimuths; and if there is any doubt remaining between 
these two, the following rules may clear it up. 

When latitude is + and declination , azimuth is between 
90 and 270; 

When latitude is + and declination +, if declination is 
greater than latitude, azimuth is not between 90 and 270 ; 

When latitude is and declination , if declination is 
greater than latitude, azimuth is between 90 and 270 ; 

When latitude is and declination +, azimuth is not 
between 90 and 270. 



NEWER NAVIGATION METHODS 115 

In other cases, and especially when latitude and declina- 
tion are nearly equal, the foregoing rules are insufficient, and 
we must consult Table 12 (p. 290), the "auxiliary azimuth 
table." This table has latitude and declination for its pair 
of arguments, the former in the left-hand vertical column, 
the latter in the topmost horizontal line : and in using the 
table it is not necessary to pay attention to the + and 
signs of latitude and declination. Start with the latitude, 
and follow its horizontal line to the right until you reach the 
column having the declination at its head. There you will 
find an "auxiliary angle," which must be compared with 
the altitude computed by formula (6), page 113. Then : 

If the computed altitude is greater than the auxiliary 
angle, and if latitude is +, azimuth is between 90 and 270 ; 

If the computed altitude is less than the auxiliary angle, 
and if latitude is , azimuth is between 90 and 270 ; 

If the computed altitude is less than the auxiliary angle, 
and if latitude is +, azimuth is not between 90 and 270 ; 

If the computed altitude is greater than the auxiliary 
angle, and if latitude is , azimuth is not between 90 and 
270. 

It will rarely happen that any of the foregoing rules will 
be needed, if the navigator will make a careful observation 
of the sun's azimuth with the azimuth circle or pelorus 
(p. 44), as soon as possible after the sextant altitude has 
been observed. The ship's course should also be specially 
recorded when this observation is made. This proceeding 
is not merely a convenience to avoid consulting the fore- 
going rules in using the azimuth table : it is really essential 
to safe navigation, for a comparison of the observed azi- 
muth with that derived from the table will make the com- 
pass error (p. 43) known. The variation is known from the 
chart ; so that if we observe the compass error, we can allow 
for the variation, and get the deviation. This can then be 
compared with the deviation table (p. 48), to see if there has 
been any change in the compass since leaving port. It is 



116 NAVIGATION 

a great advantage of the Sumner method that the sun's 
azimuth comes out as a sort of by-product, so that the com- 
pass can be verified without any additional special calcu- 
lations. 

We shall now illustrate all the above considerations by 
means of examples ; beginning with the observation already 
treated as a time-sight (p. 101). That observation we shall 
now work by the Sumner method. From page 101 we take 
the following : 

Date of observation, Dec. 18, 1917, in the afternoon; D. R. 
latitude, 42 20' N. ; D. R. longitude, 35 16' W. ; altitude observed, 
14 19' ; time by watch, 2* 29 m 58' ; C. - W., 2* 27" 8 ; chronometer 
correction, 2 m S' slow of G. M. T. ; index correction, + 4' ; height of 
eye, 24 ft. 

From the preparatory part of the calculation (p. 102), 
we also copy the following additional numbers : 

Declination, line (6), page 102 -23 24' (1) 

Greenwich apparent time (G. A. T.) of observation, 

line (8), page 102 5* 2- 35* (2) 

We have next to calculate, by the formulas on page 113, the 
altitude corresponding to the D. R. point, for which the 
latitude and longitude are given above. The longitude is 
35 16' W., or, at 15 to the hour (Table 9, p. 249) : 

D. R. longitude is 2* 21" 4* W. (3) 

Subtracting (3) from (2), according to page 112, 

gives ship's apparent time of observation, T. . 2 41 31 (4) 

We are now prepared to apply formulas (1) to (6), 
page 113. We choose formula (2) for an afternoon obser- 
vation l ; and write : 

1 For a forenoon observation we should choose formula (1), and 
should therefore need to know 24* T instead of T. This would 
make necessary another line in the form of calculation, and it would 
follow line (4). This new line might be numbered (4') ; and in it 
would be written 24* T, obtained by subtracting T (line 4) from 
24*. 



NEWER NAVIGATION METHODS 117 

Cos lat., 42 20' N. by D. R. (see Table 4, p. 238) .... 9.86879 (5) 

Cos dec., 23 24', line (1) (see Table 4, p. 219) 9.96273 (6) 

Hav. T, 2* 41 m 31', line (4) (see Table 10, p. 260) .... 9.07596 (7) 
Adding (5) to (7) gives hav. X (dropping 20, p. 25) . . 8.90748 (8) 

Now we choose formula (4), because latitude and declina- 
tion are -+- and ; 

The latitude is, by D. R 42 20' (9) 

Adding (1) and (9) according to formula (4) gives 

(lat. - dec.) 65 44' (10) 

Now we have, Table 10, page 266, No. hav. of (10) . . 0.29451 (11) 

No. hav. X, 1 line (8) 0.08082 (12) 

Adding (11) and (12), according to formula (5), page 

113, gives No. hav. Z 0.37533 (13) 

And Z, corresponding to (13) is found from Table 10, 

page 268 75 34' (14) 

Then, by formula (6) computed altitude =90 - Z (14), 

or 14 26' (15) 

This computed altitude (15) must now be compared with 
the observed altitude, fully corrected. We find : 

Obs'd alt., fully corrected, line (17), page 102, is 14 30' (16) 

Difference between (15) and (16), in minutes, is the 
distance of Sumner point from D. R. point in 
miles (p. 113). It is 4 miles (17) 

Next we must find the sun's azimuth from Table 11, page 
286. The top argument for entering the table is T, line 
(4), and it must be found in the "afternoon" lines. The 
argument for the left-hand column is the declination, line (1). 
Under T, and opposite declination, we find the tabular index 
number 5872. 2 Then we find the computed altitude, line 
(15), in the right-hand column of Table 11, page 286, and 

1 This No. hav. X comes from Table 10, page 258, without looking 
up the angle X at all. We simply find hav. X in the table, and take 
the No. hav. X out of the adjoining heavy type column. No inter- 
polations are needed, the nearest tabular numbers being sufficiently 
accurate. 

2 The index numbers and the azimuth need not be very accurate : 
it is sufficient to use the nearest tabular arguments, so that inter- 
polation is not essential. 



118 NAVIGATION 

follow its horizontal line till we again come upon the index 
number 5872. It lies about halfway between 5703 and 
5973. Going down the two columns containing these index 
numbers, we find in the afternoon azimuth lines two values 
of the azimuth, 217 and 323. The choice between these 
two numbers would be very easy, if the observer's record 
contained even a rough estimate of the sun's bearing at the 
time of observation. We have purposely not made this avail- 
able, so as to show how to consult the directions on page 
114, and there we find that when the latitude is -f and the 
declination , the azimuth is between 90 and 270. So 
we finally choose 217 for the sun's azimuth. 

Since the observed altitude (16) is greater than the com- 
puted altitude (15), the bearing of the Sumner point from 
the D. R. point, according to page 113, is the same as the sun's 
azimuth, or 217. And as we now know the bearing and 
distance of the Sumner point from the D. R. point, we can 
find its latitude and longitude by a simple application of the 
traverse table (p. 154). 

We have merely to consider the bearing and distance to 
be a course angle and distance, and imagine a ship to have 
sailed from the one point to the other. In the present case, 
the distance is 4 miles (line 17), the course 217 : and Table 1 
(p. 164) gives the corresponding latitude 3'.2, departure 2.4. 
The longitude difference is obtained from the departure by 
Table 2 (p. 174) and is, for latitude 42, about 3'.2. Drop- 
ping odd fractions, the latitude difference and longitude differ- 
ence both come out 3'. The Sumner point is therefore 3' dis- 
tant from the D. R. point in both latitude and longitude. 
And since the bearing 217 indicates on the compass card 
that the Sumner point is south and west of the D. R. point, 
it follows that : 

Lat. of Sumner point = D. R. lat. 3' = 

42 20' N. (line 9) - 3' 42 17' N. (18) 

Long, of Sumner point = D. R. long. +3' 35 19 W. (19) 

Azimuth of Sumner line (p. Ill) 307 (20) 



NEWER NAVIGATION METHODS 



119 



It is important for the reader to understand that the fore- 
going calculation is given in extended detail so as to make 
it easy for the beginner to follow. In condensed form, 
we should have the following arrangement of the calculation, 
corresponding to the condensed time-sight form (p. 104). 
Part of the work here repeated from page 104 has no attached 
reference numbers in parentheses : the new part of the work 
has references to the detailed calculation just given. 



SUMNER LINE, CONDENSED FORM. SUN 

It.: 14 19' Decl.4*: 23 23'. 7 S. 

+ 4 H. D. : 0.1 

: + 12 Decl. 4* 59 : 23 24' S. 

: - 5 Eq. time, 4*: +322.3 

It. : 14 30' H. D. : 1.2 

Eq. time, 4* 59 : +3 21.1 



Watch time : 


2 29" 58* 








C. - W. : 


2 27 8 








Chr. time : 


4 57 6 








Chr. corr'n : 


+ 28 








G. M. T. 18th : 


4 59 14 








Eq. of time : 


+ 3 21 








G. app. time : 


5 2 35 








D. R. long. : 


2 21 4W. (3) 


Ship's app. time, 


T: 2 41 31 


(4) 


hav. T (or 24* -T) 


!; 9.07596 


D. R. lat. : 


42 20' N. 


(9) 


cos lat. : 


9.86879 


Dec.: 


23 24 S. 


(D 


cos dec. : 


9.96273 








sum = hav. X : 


8.90748 








No. hav. X : 


0.08082 (12) 








No. hav. (lat. 




Lat. Dec. : 


65 44 


(10) 


dec.) : 


0.29451 (11) 


Z: 


75 34 


(14) 


No. hav. Z 


0.37533 (13) 


Comp'd alt. : 


14 26 


(15) 






Obs'd alt. : 


14 30 


(16) 






Diff.: 


4 


(17) 






Index No. : 


5872 








Azimuth : 


217 








Lat. diff. : 


3'.2 




Dep. : 


2.4 








Long. diff. : 


3'.2 


D. R. lat. : 


42 20' N. 


(9) 


D. R. long. : 


35 16' W. (3) 


Sumner pt. lat. : 


42 17 N. 


(18) 


Sumner pt. long. : 


35 19 W. (19) 


Azimuth of Sumner line : 307 


(20) 







1 See footnote, p. 116. 



120 . NAVIGATION 

When the object observed is a star (cf. p. 104) or planetj 
the choice between formulas (1) and (2), page 113, is not quite 
the same as in the case of a solar observation. We must 
use formula (1) if the star was on the east side of the sky 
when observed, which might be called a "forenoon" observa- 
tion of the star ; and we must use (2) if the star was on the 
west side of the sky, giving an "afternoon" star observa- 
tion. The use of the remaining formulas (3) to (6) is the 
same as for the sun ; but T is now no longer the ship's appar- 
ent time. Instead, it is the star's hour-angle (p. 104) ; 
to find it for use in formulas (1) and (2), and in Table 11, 
we must first calculate (p. 85) the Greenwich sidereal 
time corresponding to the G. M. T. of the observation, as 
taken from the chronometer, duly corrected for error and 
rate ; and then use the following formulas : 

(7) Greenwich hour-angle = Greenwich sidereal time right ascen- 
sion of star, 

. R . I T = Greenwich hour-angle + D. R. longitude, if east, 
\ T = Greenwich hour-angle D. R. longitude, if west. 

As an application of the Sumner method to a star observa- 
tion, let us take the observation of Sirius, Dec. 17, 1917, 
off Cape Agulhas, already treated as a time-sight (p. 105). 

From the preliminary calculations there given, we have : 

Greenwich hour-angle, line (11), page 105 2* 6 m 34* (1) 

D. R. longitude (p. 105) is 20 41' E., or by 

Table 9 (p. 249) 1 22 44 E. (2) 

By formula (8) above, we add (1) and (2), 

giving T 3 29 18 (3) 

The star bore west 1 (p. 105) so we choose formula (2) 
(p. 113), and write: 

cos lat. (p. 106, line 19), 35 20' S. by D. R. 

(see Table 4, p. 231) 9.91158 (4) 

cos dec. (p. 106, line 12), - 16 36' (Tab. 4, p. 212) 9.98151 (5) 

hav. T, 3* 29 m 18" (line 3, above) (see Table 10, p. 263) 9.28872 (6) 

Adding (4) to (6) gives, by formula (2), page 1 13, hav.Z, 9.18181 l (7) 

1 See p. 116, footnote. 

* Sum diminished by 20 (see footnote, p. 102). 



NEWER NAVIGATION METHODS 121 

Next we choose formula (3), page 113, since latitude and 
declination are both . We have : 

By formula (3), lat. - dec. = 35 20' - 16 36' = 18 44' (8) 

We now use formula (5), page 113. We have: 

No. hav. 18 44' (8) (see Table 10, p. 254) 0.02649 (9) 

No. hav. X* (7) (see Table 10, p. 261) 0.15194 (10) 

Adding (9) and (10) gives No. hav. Z 0.17843 (11) 

And Z, corresponding to (11) is found from 

Table 10, page 262 49 59' (12) 

Then, by formula (6), page 113, 

computed alt. = 90 - Z (12), or 40 1' (13) 

This computed altitude (13) must be compared 
with the observed altitude, fully corrected. 

This was (p. 106, line 18) 40 2' (14) 

Difference between (13) and (14), in minutes, or dis- 
tance of Sumner point from D. R. point in miles 
(p. 113) 1 mile (15) 

Next we find the star's azimuth from Table 11, page 287. 

The top argument for, entering the table is T, line (3), 
and it must be found in the "afternoon" lines, since the star 
bore W. The argument for the left-hand column is the 
declination, line (5). Under T (p. 287), and opposite 
declination, we find (approximately) the tabular index num- 
ber 7550. Then we find the computed altitude, 40 (13), 
in the right-hand column of the table (p. 289), and follow 
along its horizontal line until we again reach the index 
number 7550. The nearest to 7550 is 7544; and under 
this number, at the foot of the column, we find the two 
"afternoon" azimuths 260 and 280. 

These two numbers are so nearly equal that there is un- 
certainty in choosing between them. Had the observer 
taken the star's bearing by compass at the time of observa- 
tion (p. 115), the uncertainty would be removed. But 
in the absence of this information, we must have recourse 
to Table 12 (p. 290), the auxiliary azimuth table. Enter- 
ing this table with the pair of arguments of the present 

1 No. hav. here obtained from hav. without finding the angle X 
(p. 117, footnote). 



122 NAVIGATION 

problem: viz. latitude 35, declination 17, we find the 
auxiliary angle 31. The computed altitude (13) being 
40, is greater than the auxiliary angle, and the latitude is . 
Therefore, by the instructions (p. 115), the azimuth is 
not between 90 and 270. We therefore choose 280 as 
our final azimuth, since 260, the other possible value, is in 
the prohibited area between 90 and 270. 

The computed altitude (13) being less than the observed 
altitude, this observation places the Sumner point 1 mile 
(15) from the D. R. point, and bearing from it 280, the same 
as the star's azimuth (p. 113). The traverse table (p. 156) 
gives, for distance 1 and course 280, latitude 0.2, departure 
1.0. The longitude difference, by Table 2 (p. 172), is 1'.2, 
for the departure 1 .0. Therefore, since azimuth 280 indicates 
on the compass card that the Sumner point is W. and N. 
of the D. R. point, we have : 

lat. of Sumner point = - 35 20' (4) + 0'.2 = - 35 20' (16) 
long, of Sumner point = 20 41' E. (2) - 1'.2 = 20 40' E. (17) 

The bearing of the Sumner line will be 90 greater than 
the star's azimuth (p. Ill) ; so we have : 

Bearing of Sumner line = 280 + 90 = 370 ; or, 

dropping 360 = 10 (18) 

The foregoing calculation of the Sumner point from a 
star observation can of course also be put in condensed form. 
In doing so, we have repeated certain numbers from page 107 
without references in parentheses. But numbers taken 
from the extended calculation just given have their reference 
numbers attached. 

This condensed form, like the others previously given, is 
the form of calculation which would be used in actual 
navigation. It is most important, in the interest of numeri- 
cal accuracy, to make all calculations upon forms ; and no 
numbers should be written on the forms without having an 
adjoining statement as to the meaning of the numbers. 



NEWER NAVIGATION METHODS 



123 



SUMNER LINE, CONDENSED FORM. STAR 



Watch time : 16* 29" 48* 




C. - W. : - 1 23 50 




Chr. time : 15 5 58 




Chr. corr'n : - 2 28 


Obs'd alt. : 40 3' 


G. M. T. : 15 3 30 


Index : + 5 


R. A. mean sun : 17 42 10 


Table 6 : - 1 


Corr'n, past noon : 2 28 


Table 7 : - 5 


Greenw'h sid. time : 8 48 8 


Corr'd alt. : 40 2 


R. A. of Sirius : 6 41 34 




Greenw'h hour-angle : 2 6 34 




D. R. long. : 1 22 44 E. 


(2) 


T: 3 29 18 


(3) 


T or (24* - T) : 3* 29" 18* 


(3) hav. : 9.28872 (6) 


Dec. : - 16 36' 


cos : 9.98151 (5) 


D. R. lat. : - 35 20 


cos: 9.91158 (4) 


Sum of 3 = hav. X: 


9.18181 (7) 


. No. hav. X : 


0.15194 (10) 


Lat. - Dec. : 18 44' (8) ; 


No. hav. : 0.02649 (9) 


Sum of 2 = No. hav. Z : 


0.17843 (11) 


Z: 


49 59' (12) 


Computed alt. = 90 - Z : 


40 1 (13) 


Obs'd alt., corr'd : 


40 2 (14) 


Diff.: 


1 (15) 


Index No. : 7550 




Azimuth : 280 




Lat. diff. : 0'.2 Dep. : 1.0 


Long. diff. : 1'.2 


Sumner pt. lat. : - 35 20' (16) 


; long. : 20 40' E. (17) 


Bearing of Sumner line : 10 (18) 



We have now, in the foregoing examples, illustrated the 
manner of determining a Sumner line completely by ascer- 
taining the latitude and longitude of one point on the line 
(the Sumner point), and the bearing of the line itself at that 
point. It may be desired to draw the line on the chart, 
which will always interest the navigator if he is near the 
coast and has a large-scale chart. To draw it, we merely 
locate the Sumner point on the chart by its latitude and longi- 

1 See footnote, p. 116. 



124 NAVIGATION 

tude, and then draw the line through the point so that it 
will make with the meridian an angle equal to the bearing 
which has been computed for the line. The Sumner line 
should be extended in both directions from the Sumner 
point, for any convenient distance, in such a way that the 
point will be near the middle of the line. 

We can now gain a better understanding as to Sumner 
navigation by comparing the results obtained in one of the 
foregoing examples with the corresponding calculation of 
the same example as a time-sight. Thus from the same ob- 
servation (pp. 104, 119) 

As A TiME-SlGHT As A StTMNER OBSERVATION 

From D. R. latitude 42 20' N. ; From D. R. latitude 42 20' N. ; 
D. R. longitude 35 16' W., we D. R. longitude 35 16' W., we 
found the ship's longitude to be found the Sumner point to be 
35 24' W. in latitude 42 17' ; longitude 35 

19' W. ; and azimuth of Sumner 

line, 307. 

Starting with the same observed altitude, and the same 
D. R. position of the ship, we get quite different results by 
the two methods of calculation. The time-sight gives us 
nothing but a longitude ; and it will be the correct ship's 
longitude only if the D. R. latitude was also correct (p. 101). 
Therefore the time-sight calculation leaves us with both 
latitude and longitude still affected by possible errors in the 
D. R. latitude. 

On the other hand, the Sumner calculation gives us both 
a latitude and a longitude, but neither belongs to the ship's 
position. They both belong to the position of the Sumner 
point, but they are free from the effects of any D. R. errors. 
They fix the Sumner point only, but they fix it correctly. 
Furthermore, our knowledge that the ship is somewhere 
on the Sumner line is also a fact, free from error. So what 
we learn from the Sumner method is sure ; what we get by 
the older methods is all really D. R. information in some 



NEWER NAVIGATION METHODS 125 

degree. The Sumner method is independent of D. R., an 
advantage of which the value cannot be estimated too highly. 

Furthermore, it can be shown mathematically (cf. p. Ill) 
that a single observation can never really do more than 
determine a line on which the ship must be. Even a noon- 
sight does no more than this ; for in determining the ship's 
latitude, it really only makes known a horizontal line (the 
ship's latitude parallel) on the chart. In other words, for 
a noon-sight the Sumner line is horizontal, or has a bearing 
of 90. And it will always come out 90, if a noon-sight is 
worked as a Sumner observation. 

But the principal purpose of our present comparison of 
the two methods of calculation is to warn the navigator 
against falling into the error of imagining the ship to be at 
the Sumner point. The observation does no more than tell 
us where the Sumner point is, and that the ship is somewhere 
on the line ; so far as the observation is concerned, all points 
on the line are equally likely to be the ship's true position. 
Therefore it is misleading to call the Sumner point the ship's 
"most probable position." Were it so, a second observation, 
made later in the day, would give another "most probable 
position" of the ship. We should then be naturally led to 
take as the ship's final location a point midway between the 
two "most probables," ascribing their divergence to possible 
errors of observation. But the ship's real position we already 
know (p. Ill) to be at the intersection of the two Sumner 
lines resulting from the two observations. And this inter- 
secting point may be many miles from both "most proba- 
bles," and from the above-mentioned midpoint between 
them. 

Less than two observations cannot fix the ship's position 
completely; when two have been made, a correct applica- 
tion of the Sumner method requires that the intersection 
point of two Sumner lines be determined by calculation. 
But before explaining the method of doing this, we must 
describe an excellent alternative way of making Sumner 



126 NAVIGATION 

calculations such as we have given in the above examples. 
The results are the same results as before, but they are 
obtained with less work, and quite without logarithms, by 
means of special tables such as our Table 13 (p. 292), 1 which 
we shall call Kelvin's Sumner Line Table. 

This table has a pair of arguments (p. 11), a and 6, a ap- 
pearing at the heads of the tabular columns, and b in the 
left-hand column of each page. Corresponding to these 
two arguments, the table gives two angles, K and Q ; so that 
whenever a and b are given we can find the corresponding 
K and Q ; or, if a and K should be given, we can find the 
corresponding 6 and Q. 

In the Sumner problem we obtain, by preparatory calcu- 
lation (cf. pp. 119, 123), the following data: 

Declination of sun (or star) ; D. R. latitude ; D. R. longitude ; 
T, the ship's apparent time of the observation for the sun, or the 
hour-angle for a star ; 

and we wish to get the computed altitude and the azimuth. 

The principle on which Table 13 depends is that the D. R. 
latitude and longitude being always somewhat uncertain, 
we can, if we choose, change them by reasonable amounts 
before beginning our calculations. The Sumner point will 
then be determined by its distance and bearing from the 
changed D. R. point, instead of the original D. R. point. 
By this device the tabular calculation is much facilitated. 
The use of the table is easy after a little practice, the work 
being divided into a series of separate operations. In de- 
scribing these operations we have used small subscript num- 
bers, to distinguish the several arguments, etc. ; as, for in- 
stance, in Operation 1 we use a\, b\, Ki. 

1 These tables were first published by Lord Kelvin in 1876. 
More extended ones were recently issued by Lieutenant de Aquino, 
of the Brazilian Navy; and these were reprinted by the Hydro- 
graphic Office, United States Navy, in 1917. Aquino also improved 
Kelvin's method of using his table. 



NEWER NAVIGATION METHODS 127 

OPERATION 1, requiring no interpolation. Enter Table 13 
with : 

Arg. ai = declination, taken without regard to + or sign, and cor- 
rect to the nearest whole degree only ; 
Arg. 61 = T, if T is between 0* and 6* ; 

= 12* - T, if T is between 6* and 12* ; 
= T - 12*, if T is between 12* and 18*; 
= 24* - T, if T is between 18* and 24*; 

and before use 61 must be turned into degrees with 
Table 9 (p. 249). It need be correct to the nearest 
degree only. This proceeding will make fei always 
less than 90. 

Then take from the table the tabular angle KI, also correct 
to the nearest degree only. 

OPERATION 2, requiring simple interpolation. Enter the 
table a second time with : 

Arg. o = the KI, obtained in Operation 1. 

Then, under this a?, run down the ^C-column until you 
find the declination (taken without regard to + or sign) ; 
so that, in other words, K 2 = declination. 

Take from the table the angle Q 2 , which stands next to 
the declination K z , and also the & 2 , which is in the left-hand 
argument column, in the same horizontal line with the 
declination K 2 in the /f-column. It will rarely be possible 
to find the declination (which must this time be exact to 
the nearest minute) in the K-column; so that a simple 
interpolation will be necessary in getting $2 and 6 2 . An 
example of this interpolation will be found on page 129 ; and, 
as we shall see, it is practically the only numerical calculation 
required in the whole problem. The Kelvin method is very 
much shorter than it looks. 

The angle Q 2 is used in choosing the longitude of the 
"changed D. R. point"; the latitude of that point will be 
found in Operation 3. To utilize Q 2 for a sun observation, 
calculate the Greenwich apparent time (G. A. T.) of the 



128 NAVIGATION 

observation, as on page 102, line (8), and turn it into de- 
grees with Table 9 (page 249). Then : 

(1) W. long, of changed D. R. point = G. A. T. - Q 2 , if, in Oper- 

ation 1, T was less than 6*; 

(2) W. long, of changed D. R. point = G. A. T. - (180 - Q 2 ) if, 

in Operation 1, T was between 6* and 12*; 

(3) W. long, of changed D. R. point = G. A. T. - (180 + Q 2 ) if, 

in Operation 1, T was between 12* and 18*; 

(4) W. long, of changed D. R. point = G. A. T. - (360 - Q 2 ) if, 

in Operation 1, T was between 18* and 24*. 

When the subtractions in these formulas cannot be made, 
the G. A. T. may be increased by 360 ; and when the west 
longitude comes out greater than 180, subtract it from 360, 
and call it east longitude. 

In the case of a star, we must use, in the above formulas, 
the Greenwich hour-angle, instead of the G. A. T. See 
page 105, line (11), for the method of obtaining it. 

OPERATION 3, requiring no interpolation. Enter the table 
a third time with : 

Arg. o 8 = Ki, again as obtained in Operation 1. 

(5) Arg. b s = 90 - (b 2 + changed D. R. lat.), if latitude and 

declination are of opposite signs, one + and 
one ; 

(6) Arg. fc s = (b t + changed D. R. lat.) - 90, if T was between 

90 and 270; 

(7) Arg. 6, = 90 - (6 2 - changed D. R. lat,), if latitude is less 

than 62; 

(8) Arg. &, = 90 + (& 2 - changed D. R. lat.), if latitude is 

greater than &. 

In choosing among formulas (5) to (8), give them pre- 
cedence in order ; do not use (7) or (8) if the conditions 
stated for (5) or (6) are satisfied. And at this point, use 
your privilege of choosing any reasonable changed D. R. lati- 
tude for the ship ; and choose one that differs as little as pos- 
sible from the original D. R. latitude, and that yet makes 
6 3 a whole number of degrees. In this way, all further 



NEWER NAVIGATION METHODS 129 

interpolation is avoided. Having once chosen among the 
formulas, the latitude is used without regard to + or 
signs. 

To complete Operation 3, having entered the table with 
the pair of arguments a 3 and & 3 , take out the tabular K 3 
and Q 3 . 

K 3 is now the computed altitude, to be used (p. 113) in 
locating the Sumner. point from the changed D. R. point; 
and Qs is the sun's true azimuth, which will always come 
from the table less than 90. If the ship is in the northern 
hemisphere, this azimuth must be counted from the north 
point of the horizon if, in Operation 3, we used formulas (6) 
or (7) ; or from the south point of the horizon, if we used 
formulas (5) or (8). With the ship in the southern hemi- 
sphere, interchange the north and south points of the horizon 
in these directions. And in both hemispheres, the azimuth 
will of course be counted toward the east or west, according 
as the observation was a "forenoon" or "afternoon" one 
(cf. p. 120). 

We shall now use Table 13 for the example given on page 
119 in condensed form. We have (p. 127) : 

OPERATION 1. 

a\ = dec. = 23, p. 119, line (1), to the nearest degree; 

&! = T = 2 h 41-" 31', p. 119, line (4) = 40, to the nearest 
degree ; and, with ai and bi as arguments, Table 13 gives 
(p. 298) : KI = 36, to the nearest degree. 

OPERATION 2. 

02 = K! = 36. 
K z = 23 24', p. 119, line (1) 

and, with 02 and K 2 , we must find Q 2 and 6 2 . Running down 
the column headed a = 36 (p. 302), we find : 

When K 2 = 23 5', Q 2 = 39 43', b 2 = 29, 
When K 2 = 23 51', Q 2 = 40 0', b 2 = 30. 

We wish to interpolate for K 2 = 23 24', which is 19' 
down from 23 5' toward 23 51'. The whole distance from 



130 NAVIGATION 

23 5' to 23 51' is 46'. Therefore we must interpolate 
down f of the whole interval from Q 2 = 39 43' to Q 2 = 
40 0'. The difference between these two Q 2 's is 17' ; there- 
fore the final Q 2 , belonging to K 2 = 23 24', is 39 43' + 
^ X 17' = 39 43' + 7' = 39 50'. Similarly, the difference 
between the two 6 2 's being 60', the final value of 6 2 , for 
K 2 = 23 24', is 29 + if X 60' = 29 25'. These two 
little interpolations are practically all the calculation required 
in the whole problem. 

To find the longitude of the changed D. R. point from the 
above Q 2 = 39 50', we take from page 102, line (8), 

Greenwich apparent time of observation, 5* 2 m 35* 

which, by Table 9 (p. 249) is, 75 39' 

We now use formula (1), page 128, because T, in Opera- 
tion 1, was less than 6 A . We get : 

W. long, of ch'd D. R. pt. = G. A. T. - Q, = 75 39' - 39 50' 
= 35 49' W. 

OPERATION 3. 

03 = Ki = 36. 

The D. R. latitude is + 42 20' (p. 119, line (9)) ; and as 
the declination is , we choose formula (5), page 128. 
This, without changing the D. R. latitude, would give 6 3 = 
90-(&2+D. R.lat.) =90 -(29 25'+ 42 20') = 90- 71 45'; 
but by choosing a changed D. R. latitude of 42 35', we shall 
make 6 3 a whole number of degrees. So we have : 
6 3 = 90 - (62+ changed D. R. latitude) = 90 - (29 25' 
+ 42 35') = 90 - 72 = 18. 

Now we enter the table with the arguments a 3 = 36, and 
63 = 18, and obtain, without interpolation (p. 302) : 

K> = computed altitude = 14 29', 
Qt = sun's true azimuth = 37 22'. 

This azimuth must be counted from the south point of 
the horizon, since we used formula (5) in Operation 3 ; and 



NEWER NAVIGATION METHODS 131 

as the observation was an afternoon one, the correct azi- 
muth will be S. 37 22' W. (cf. p. 19). Counted in the United 
States Navy way, from the north toward the east, and so 
around to 360, the azimuth will be 217 22'. 

On page 119, we found : Computed altitude, 14 26'; azi- 
muth, 217. 

This computed altitude differs by 3' from the value just 
found by Table 13. The difference is due to our having 
changed the D. R. point. 

From the changed D. R. point, in latitude 42 35' N. ; 
longitude 35 49' W., we now calculate (see Condensed Form, 
next page) the position of the Sumner point to be : latitude 
42 34' N. ; longitude 35 50' W. The former position, as 
obtained on page 119, was : latitude 42 17' N. ; longitude 
35 19' W. 

These two Sumner point positions should lie on the 
same Sumner line if the method of Table 13 gives correct 
results; and they will satisfy this test, if the bearing 
of a line joining them agrees with the azimuth of the 
Sumner line, which is 217 + 90 = 307. From the two 
Sumner point positions we have : latitude difference = 17' ; 
longitude difference = 31'; departure (Table 2, p. 174) 
= 23.0. The traverse table (p. 164) gives, for latitude 17, 
departure 23.0, the distance 28, course 307. The agree- 
ment is perfect, and shows that the same Sumner line 
passes through both points, though they are 28 miles 
apart. This test also shows that the calculation may 
indicate any point on the Sumner line as the Sumner point, 
if the D. R. position of the ship is uncertain : and so 
we again call attention to the error of taking the cal- 
culated Sumner point as the ship's most probable position 
(cf. p. 125). 

We now, as usual, repeat the above calculation by Table 13, 
in condensed form, and including the final determination 
of the position of the Sumner point from the changed D. R. 
point. 



132 NAVIGATION 






SUMNER LINE BY TABLE 13, CONDENSED FORM. SUN 
[The following is taken from page 119.] 



Decl., 4* : 


- 23 23'.7 




Eq 


. of time : + 3 


' 22.3 


H. D. : 


0.1 




H. 


D. : 


1.2 


Decl., 4*59: 


-23 24 




Eq 


. time : + 3 


21.1 


Watch time : 


2* 29* 


58* 




Obs'd alt. : 


14 19' 


C. -W.: 


2 27 


8 




Index : 


+ 4 


Chr. time : 


4 57 


6 




Table 6 : 


+ 12 


Chr. corr'n : 


+ 2 


8 




Table 7 : 


-5 


G. M. T. : 


4 59 


14 




Corr'd alt. : 


14 30 


Eq. of time : 


+ 3 


21 




D. R. lat. : 


42 20' N. 


G. app. time : 


5 2 


35 




D. R. long. : 


35 16' W. 


D. R. long. : 


2 21 


4 W. 


(3) 






Ship's app. time, 


T: 2 41 


31 


(4) 







[The following is calculated with Table 13.] 

OPERATION 1 OPERATION 2 

ai = dec. =23 at = K i = 36 

bi = T = 2^ 41" 31(4) Ki = dec. = 23 24' 

= 40 Table 13, Q t = 39 50' 

Table 13, Ki = 36 Table 13, bt = 29 25' 

Greenwich app. time = 5* 2> 35* = 75 39' 
By page 128, form. (1), W. long, of changed D. R. pt. = G. A. T. - Q, 

= 35 49' W. 
Lat. of changed D. R. pt. = 42 35' N. 

OPERATION 3 
a. = Ki = 36 

bi = 90 - (6, + changed D. R. lat.) = 18 
Table 13, K* = comp'd alt. - 14 29' 

Table 13, Q t = azimuth of sun = 37 22' 

or, by U. S. Navy = 217 22' 

Azimuth of Sumner line = 217 22' + 90 

= 307 22' 

Dist. of Sumner pt. from changed 

D. R. pt. = corr'd obs'd alt. comp'd alt. = 1' or 1 mile 
Bearing of Sumner pt. from changed D. R. pt. = 217, 
since comp'd alt. is less than obs'd alt. 

Dist. 1, on course 217, gives lat. diff., 0'.8; dep., 0.6; long, diff., 0'.8 
Lat. of Sumner pt. = lat. of ch'd D. R. pt. - lat. diff. = 42 34' N. 
Long, of Sumner pt. = long, of ch'd D. R. pt. + long. diff. = 35 50' W. 

A practised navigator can make the above complete calcu- 
lation in a few minutes, as there are no logs used ; and any 
one can easily obtain the necessary practice at sea by simply 
forming the habit of working his sights both as time-sights 
and as Sumners. To illustrate the subject further, we now 
give, in condensed form, the Star Example of p. 123, worked 
by Table 13. 



NEWER NAVIGATION METHODS 133 

SUMNER LINE BY TABLE 13, CONDENSED FORM. STAR 
[The following is taken from page 123.] 

Watch time: 16* 29 48 Obs'd alt. : 40 3' 

C. - W. : - 1 23 50 Index : + 5 
Chr. time : 15 5 58 Table 6 : - 1 
Chr. corr'n : - 2 28 Table 7 : - 5 
G. M. T. : 15 3 30 Corr'd obs'd alt. : 40 2 
R. A. mean sun : 17 42 10 

Corr'n, past noon : 2 28 Dec. of Sirius : 16 36 

Greenwich sid. time : 8 48 8 D. R. lat. : - 35 20 

R. A. of Sirius : 6 41 34 

Green, hour-angle : 2 6 34 

D. R. long. : 1 22 44 E. 
T: 3 29 18 

[The following is calculated with Table 13.] 

OPERATION 1 OPERATION 2 

01 = dec. =17 oi = Ki = 49 

61 = T = 3* 29" 18* Kt = dec. = 16 36' 

= 52 Table 13, Q, - 51 57' 

Table 13, Ki = 49 Table 13, bt = 25 49' 

By page 128, form. (1), 

W. long, of changed D. R. pt. = Green, hour-angle Qz 1 

339 41' 
20 19' E. 
Lat. of changed D. R. pt. = - 35 49' 

OPERATION 3 

at = Ki = 49 

By form. (8), page 128, 6. = 90 + (61 - changed D. R. lat.) = 80 

Table 13, Ki = comp'd alt. = 40 15' 

Table 13, Q = az. of Sirius = N. 81 25' W. 

or, by U. S. Navy = 278 35' 

Az. of Sumner line = 368 35', or 8 35' 

Dist. of Sumner pt. from changed 

D. R. pt. = corr'd obs'd alt. comp'd alt. = 13' or 13 miles 

Bearing of Sumner pt. from changed D. R. pt. = 99, 

since comp'd alt. is greater than obs'd alt. 

Dist. 13, on course 99, gives lat. diff ., 2'.0 ; dep., 12.8 ; long, diff ., 15'.9 

Lat. of Sumner pt. = lat. of ch'd D. R. pt. + lat. diff. = - 35 51' 

Long, of Sumner pt. = long, of ch'd D. R. pt. + long. diff. = 20 35' E. 

To complete this part of our subject, it remains to show 
how the position of the ship can be found at the intersec- 
tion of two Sumner lines (pp. Ill, 125) resulting from 
two different observations. Figure 18 explains the nature of 
the problem ; and it is almost exactly the same figure and 

1 Qz being larger than the Greenwich hour-angle, the latter was 
increased by 360, to make the subtraction possible (p. 128). 



134 



NAVIGATION 



problem treated in Chapter V, when we discussed fixing a 

ship's position by means of "bearings from the bow" 

(p. 54). 
The two Sumner lines in Fig. 18 are SL and S'L, passing 

through the two Sumner points S and S f , whose latitudes 

and longitudes are known 
by calculation from the 
observed altitudes. The 
bearings or azimuths of the 
two Sumner lines from the 
north are the two angles 
NSL and N'S'L, which are 
also known from the pre- 
vious calculations. It is 
now required to find the 
latitude and longitude of 
the intersection point L, 
where the ship is situated. 
The similarity of this 
problem to the former one 
/^ in Chapter V becomes plain, 

FIG. ^.-Intersection of Sumner Lines. if WG ima S ine a SeCOnd shi P 

sailing from one Sumner 

point to the other, as from S to S', and taking bearings 
from her bow upon our ship, located at L. These bearings 
will be the two angles S'SL and S"S'L. If the second 
of these angles should happen to be just twice as big 
as the first, the distance S'L between the two ships at 
the time of the second bearing would be equal (p. 54) to 
the distance SS' run by the imagined ship between the two 
observations. 

This would enable us to fix the position of the imagined 
ship at S', if L were a lighthouse ashore. But if L is our 
ship, and S' a Sumner point of known position, the same 
observations of bow bearings would fix the position of our 
ship at L. Nor is it necessary (or possible) to measure 




NEWER NAVIGATION METHODS 135 

such imaginary bearings, or read the patent log to get the 
distance run by an imagined ship. 

For the distance and bearing of the second Sumner point 
from the first can be obtained from their known latitudes 
and longitudes with the traverse table. Thus the line SS' 
(marked "distance") and the bearing (or course) angle 
NSS' become known. Furthermore, the "bow bearing" at 
S is the angle S'SL, and it is equal to the difference NSL 
NSS'. We have just seen that NSS' is obtained from the 
traverse table ; and NSL is the calculated azimuth of the 
Sumner line through S. In a similar way we get the other 
"bow bearing" S"S'L. If this were twice the first one, the 
"required distance" S'L in the figure would be equal to the 
known distance SS' between the two Sumner points. If 
not, it can be easily shown mathematically that : 

(1) Required distance = known distance X a factor, 

(2) log factor = sin S'SL - sin (S"S'L - S'SL). 

By these simple formulas the required distance S'L might 
be found : and as we also know the latitude and longitude 
of the Sumner point S', and the azimuth or bearing of S'L, 
the traverse table will make known the latitude and longi- 
tude of the ship at L. It is to be noted also that as we are 
at liberty to call either of the Sumner points S', it is desirable 
to call that one S' which has the larger "bow bearing," 
so that there will be no difficulty about subtracting S'SL 
from S"S'L. 

The factor of formula (2) above can practically always 
be found in our Table 14, the Sumner Intersection Table, 
without using logarithms. The pair of arguments of the 
table are the smaller "bow bearing" and the larger "bow 
bearing"; the tabular number is the factor of formula (1) 
above, and will always give the distance of the intersection 
point from that one of the two Sumner points for which 
the bow bearing was the larger. 

And it should not be forgotten that the Sumner line really 



136 NAVIGATION 

extends equally in both directions (p. 124) from the Sumner 
point, whereas, in Fig. 18, we have extended it mainly 
in the direction of the intersection point L. Now the cal- 
culated azimuth of any Sumner line may be changed 180 
at will, because the bearings of the two ends of the line from 
the Sumner point differ by 180, and we may take the bear- 
ing of the line to be the bearing of either end from the Sumner 
point in the middle of the line. Figure 18 shows, however, 
that for the purpose of the present problem we must choose 
the bearing of that end of the line which is nearest the point 
of intersection L; nor does the choice ever offer difficulty, 
because the known D. R. position of the ship at L, when 
compared with the known positions of the two Sumner 
points, will always indicate whether L bears east or west 
of either Sumner point, and also whether it bears north or 
south. And the bearing of L once chosen, we can always 
find either of the two bow bearings by this formula : 

(3) Bow bearing = bearing of Sumner line minus bearing 
of the second Sumner point S' from the first point S. 

In using formula (3) it is allowable to increase the bear- 
ings of the Sumner lines by 360, when necessary to make 
the subtractions possible, and if the formula brings out bow 
bearings larger than 180, subtract them from 360, and 
proceed as before. 

It is also always desirable to draw a rough sketch for 
every intersection problem occurring on shipboard so as to 
guard against accidental large errors like 90 or 180 in ob- 
taining the two bow bearings; and also to make sure that 
the latitude and longitude of the intersection point L are 
correctly computed with the traverse table. 

The foregoing assumes that the ship did not move from 
the point L between the two sextant observations from which 
the two Sumner lines were calculated. This will rarely 
be the case, because it is very desirable that the two observa- 
tions, if they are both sun observations, be separated by 



NEWER NAVIGATION METHODS 137 

three or four hours, if possible. The condition of an unmov- 
ing ship will occur only if she is a sailing vessel becalmed, 
or a steamer at anchor ; or if the two observations are made 
at nearly the same time upon two different heavenly bodies, 
such as two stars. 

High accuracy in the resulting "fix" (p. 53) of the ship 
will then be attained, if the azimuths of the two stars differ 
by about 90 at the time of observation. The same favor- 
able condition will be secured if one of the observations is 
made upon a star near upper transit (pp. 89, 96), in the 
twilight just before sunrise or after sunset; and the other 
observation, at nearly the same time, upon the sun, when 
it is about 12 or 15 above the horizon. 

But if the ship has traveled a considerable distance between 
the two observations, it is necessary to allow for such travel 
before calculating the intersection point. Suppose she has 
gone a distance D, upon a course C, by D. R., between the 
two observations. Then simply find from Tables 1 and 2 
the difference of latitude and longitude corresponding to 
distance D and course C and apply them as corrections to 
the latitude and longitude of the Sumner point belonging 
to the first observation. Everything else, including the 
bearing of the first Sumner line, remaining unchanged, 
the calculation then proceeds by Table 14, just as if the 
ship had not moved. The computed intersection point is 
then the ship's position at the time, of the second sextant 
observation. 

We shall now work some intersection examples. 

Suppose we have two Sumner lines, as shown in the rough 
sketch, Fig. 19, taken on board a ship becalmed. The 
two sextant observations give : 

FOR ONE SUMNER POINT, S FOR THE OTHER POINT, S' 



lat. 1 

long. 

bearing of Sumner line 



4234'N. 42 50' N. 

35 50' W. 35 36' W. 

307 93 (changed to 273) 



1 As found on page 132. 



138 



NAVIGATION 



The rough sketch, Fig. 19, having been made, and the 
two "bow bearings" marked with little circular arcs as 
shown, we call that one of the two Sumner points S', which 
has the larger bow bearing ; and, for the point S', we change 




FIG. 19. Rough Sketch of Sumner Intersection. 

the bearing of the Sumner line from 93 to 180 + 93 = 
273, so as to count the bearing for that end of the line which 
is toward the intersection point L (p. 136). The other 
bearing, 307, for the point S, is already correctly counted. 

We now have, from the two Sumner point latitudes and 
longitudes : latitude difference =^ 16' ; longitude difference = 
14' ; departure (Table 2, p. 174, for middle latitude 43) = 
10.2 ; and, for latitude difference = 16, departure = 10.2, 
we find (Table 1, p. 162), distance = 19, course = 32. The 
distance between the two Sumner points is therefore 19 
miles, and the bearing of S' from S is 32. 

Now we apply formula -(3), page 136, and find : 

Smaller bow bearing at S = 307 - 32 = 275. 
Larger bow bearing at S' = 273 - 32 = 241. 

Being larger than 180, these must be subtracted from 
360 (p. 136), giving : 

Smaller bow bearing = 85; Larger bow bearing = 119. 

Next we refer to Table 14, and find with the smaller 
bearing 85, and the larger 119 the factor 1.78 (p. 322). 



NEWER NAVIGATION METHODS 139 

According to formula (1), page 135, we then have: 
Required distance LS' = distance SS' X factor 
= 19 X 1.78 = 33.8 miles. 

Therefore the position of the ship at L is distant 33.8 
miles from AS', and she bears 273. With this distance and 
bearing or course angle, the traverse table (p. 154) gives : 
latitude = 1.8, departure = 33.8. For the departure 33.8, 
Table 2 gives, for the middle latitude 43 (p. 174), differ- 
ence longitude = 46'.2. The bearing 273 showing that the 
intersection point L is N. and W. of S f , we have : 

Latitude of ship at L = 42 50' N. + 1'.8 = 42 51'.8 N. 
Longitude of ship at L = 35 36' W. + 46'.2 = 36 22' W. 

As a second example take the following two Sumner lines, 
as shown in the rough sketch, Fig. 20. The two sextant 
observations give : 

FOR ONE SUMNER POINT, S FOR THE OTHER POINT, S' 

lat. : 14 26' N. 15 30' N. 

. long. : 77 8' W. 76 22'. 5 W. 

bearing of line : 53 135 

And suppose the ship, in the interval 
between the two sextant observations, has 
traveled a distance D = 31 miles, on course 
C = 205. We must begin (p. 137) by 
shifting the first Sumner point S a dis- 
tance D, on the course C. For this course 
and distance, we have (Table 1, p. 160) : 
lat., 28M; dep., 13.1; diff. long., 13'.5 FIG. 20. Rough 

(Table 2, p. 168). Sketch^ Sumner 

Therefore, the latitude and longitude of 
the first Sumner point must be corrected (p. 137) as follows : 

For the point S, lat. = 14 26' N. - 28M = 13 58' N. 

long. = 77 8' W. + 13'.5 = 77 21'. 5 W. 

Bearing (unchanged) = 53. 
We now have, for the two Sumner points : lat. diff., 92' ; 




140 NAVIGATION 

long, diff., 59' ; dep., 57.0 (p. 169) ; dist., 108 miles (p. 162) ; 
bearing of S f from S, 32. 

Now we have, by formula (3), page 136 : 

Smaller bow bearing at S = 53 - 32 = 21. 
Larger bow bearing at S' = 135 - 32 = 103. 

Table 14 (p. 319) gives the factor 0.36 ; so that the ship at 
L is distant from S' 108 X .36 = 38.9 miles, and bears 135. 
For this distance and bearing we have (Table 1, p. 166), 
latitude = 27'.6; departure = 27.6; and longitude differ- 
ence (Table 2, p. 168) = 28'.6. Finally, then, at the time 
of the second sextant observation, the ship at L was in 
latitude 15 30' N. - 27'.6 = 15 2'.4 N. ; and in longitude 
76 22'.5 W. - 28'.6 = 75 54' W. 



CHAPTER X 
A NAVIGATOR'S DAY AT SEA 

THE present chapter contains a number of examples by 
means of which the reader can gain facility in the use of the 
methods set forth in the preceding pages. 

The steam yacht Nav is bound from New York to 
Colon, and the captain plans to take his departure from 
the Sandy Hook Lightship, on Dec. 18, 1917, as early as 
possible in the morning. 

The first bit of navigation, to be accomplished before the 
yacht leaves her anchorage in the "Horseshoe," is to ascer- 
tain by D. R. methods the proper course to steer from 
Sandy Hook. A glance at the track chart of the north 
Atlantic shows that she must go by way of Crooked Island 
Passage, and the Windward Passage between Cuba and 
Haiti. It is also apparent from the chart that the first land 
to be sighted among the islands is Watlings Island, and that 
the proper course should pass to the eastward of it. 

The position of Sandy Hook Lightship l is lat. 40 28' N. ; 
long. 73 50' W. Hinchinbroke Rock, at the southern end 
of Watlings Island, is in lat. 23 57' N. ; long. 74 28' W. 
But the course should be shaped for a point about 12 miles 
east of Watlings Island, to be perfectly safe. The position 
of such a point is (approximately) lat. 23 57' N. ; long. 74 
15' W. 2 

1 There is an excellent list of latitudes and longitudes in Bow- 
ditch's " Navigator." 

2 The difference between this longitude and that of Hinchinbroke 
Rock is 13' ; but 13' here corresponds to about 12 miles, on account 
of Table 2. 

141 



142 NAVIGATION 

ABSTRACT OF LOG. Steam Yacht Nav, Dec. 18, 1917 







PATENT 
Loo 


COMPASS 
COUKSB 


TRUE 
COURSE 


7 : 02 A.M. 


Took departure from Sandy 
Hook Lightship 


26.2 


S. 


188 


7:21 


Sunrise, observed azimuth 


31.0 


S. 


188 


8:00 




41.0 


S. 


188 


9:00 




57.2 


S. 


188 


9:36 
9:42 


Bow bearing, Barnegat .... 
Altitude and azimuth 


67.0 
69.1 


S. 

S. 


188 
188 


9:57 


Beam bearing, Barnegat . . . 


72.5 


S. 


188 




(fix, lat. 39 45' N. ; long. 










73 59' W.) 








10:00 
10:07 


Changed course 


73.4 
75.3 


S. 
S.^E. 


188 
182 


11:00 




88.7 


S.JE. 


182 


11:42 


Ex-mer. obs'n lat. 39 19'; 
D. R. long. 73 58' 


98.5 


S.iE. 


182 


12:00 




102.6 


*S* -_ 1 

S.E. 


182 


1 : 00 P.M. 




117.7 


S.JE. 


182 


2:00 




133.0 


S.JE. 


182 


3:00 




149.0 


S.iE. 


182 


4:00 




163.8 


S.JE. 


182 


4:12 


Alt. and az., fix, lat. 38 11' ; 
long. 73 54' 


166.9 


S.iE. 


182 


5:00 




182.0 


S.iE. 


182 


6:00 




197.2 


S.fE. 


182| 



By the method of page 20, the course from Sandy Hook 
Lightship should be 181, and the distance is 990 miles. 
These numbers, and all subsequent numbers in the present 
chapter, should be verified by the reader. 

The distance being quite large, it is well to check it by 
the logarithmic method, page 33. The result by this method 
is: course 181 14', distance 991.7 miles. 

The chart also shows that this course will carry the yacht 
very near Barnegat Light, on the coast of New Jersey. The 
position of this light is lat. 39 46' N. ; long. 74 6' W. The 
captain decides that it will be well to plan passing this light 



A NAVIGATOR'S DAY AT SEA 143 

at about 5 miles' distance. The position of a point 5 miles 
east of Barnegat Light is lat. 39 46' N., long. 73 59' W. The 
course and distance to this point from Sandy Hook Ship 
are 189 and 42.5 miles. This course is so nearly the same 
as the course to Watlings Island that the captain decides 
to steer the 189 course. 

All this work must be complete before reaching Sandy 
Hook, for the course from the lightship must be ready for 
the quartermaster before the lightship is passed. And 
there is still more preliminary work. For the courses cal- 
culated above are true courses (p. 43) and the quarter- 
master must have the compass course, so that he may be 
able to steer the yacht. The method of calculating the 
compass course from the true course is given on page 48 ; and 
in applying it the captain must have his deviation tables 
at hand. We shall assume that the tables printed on pages 48 
and 49 were the ones furnished by the compass adjuster for 
the present voyage. 

An examination of the Atlantic track chart shows that in 
the vicinity of Sandy Hook, the variation, V, is 10 W., or 
10. By formula (3) (p. 49), we then have, since the true 
course T is 189 : 

Magnetic course = M = T - V = 189 - (- 10) = 199. 

The second deviation table (p. 49) shows that when the 
magnetic course (or magnetic bearing of ship's head) is 199, 
the deviation, D, is + 18. Then, with V = - 10, D = 18, 
formula (1), page 45, gives : 

Compass error = E = V + D = - 10 + 18 = + 8. 

And from formula (2), page 45 : 

Compass course C = T - E = 189 - 8 = 181 ; 

and so the yacht must be steered on a 181 compass course 
for Barnegat. But the quartermaster is to steer by " points " 
so that the course nearest the 181 course is due south. The 
captain decides to have the yacht steered due south by 



144 NAVIGATION 

compass, and is prepared to give the quartermaster his 
orders as soon as Sandy Hook Lightship shall be reached. 

The foregoing preliminary work having been completed 
the previous day, the anchor is tripped at the Horseshoe 
about an hour before daylight on Dec. 18, the weather being 
fine, sea smooth, and wind light from the northwest. The 
lightship is reached and passed at 7 : 02 A.M., ship's time, civil 
reckoning, the ship then taking her departure. At that 
moment, the patent log is read, and found to register 26.2 
miles. The quartermaster gets his orders to steer south; 
and all the above facts are duly recorded in the log-book. 
And at every hour thereafter, 8, 9, 10, etc., a similar record 
must be made in the log-book. 

The next event is sunrise, which occurs at 7 : 21, very 
soon after leaving the lightship. The sun's compass bearing 
can then be very conveniently observed, and will furnish 
an excellent check on the compass adjuster. This observa- 
tion was made at 7 : 21 A.M., ship's time, civil reckoning, 
corresponding to 19* 21 m , Dec. 17, ship's apparent time, 
astronomic reckoning; and the sun's bearing or azimuth 
was 113 by compass. This was entered in the log-book, 
and at the same time the patent log was read, and found to 
be 31.0 miles. 

To check the deviation table, the procedure was then as 
follows : 

By patent log the yacht had proceeded from the light- 
ship a distance of 31.0 26.2 = 4.8 miles, on a compass 
course of 180, or true course of 188; by D. R., she had 
therefore reached the position lat. 40 23' N. ; long. 73 51' W. 
The sun's declination, from the almanac, is 23 23', and 
the (approximate 1 ) T (p. 100) is 19* 21 m . The sun's true 
azimuth is found from Table 11 to be 121 ; and in using the 
table for this purpose take the altitude of the sun, for the 

1 If there is any chance of this T being much in error, the cap- 
tain's watch, by which the observation is timed, must be compared 
with the chronometer. See p. 94. 



A NAVIGATOR'S DAY AT SEA 145 

moment of sunrise, to be 0. The observed compass azi- 
muth having been 113, formula (2), page 45, gave E = T-C 
= 121 - 113 = +8. Then from formula (1), page 45, 
j) = E -V = +8- (- 10) = + 18. As expected, this 
deviation agrees with the deviation table, which would 
not be likely to go wrong so soon after the beginning of a 
voyage. 

At 8 A.M. the patent log read 41.0; and at 9 A.M., 57.2. 
The course was still S. by compass, or 188, true course. 

At 9 : 24 Barnegat Light was sighted by the lookout, and 
the mate was ordered to take bow-and-beam bearings (p. 55) 
upon it. 

At 9 : 36, the light bore 225 by compass, or 45 from the 
bow ; patent log, 67.0. 

At 9* 42 m 28' by his watch the captain took the altitude 
of the sun's lower limb with the sextant, and found it to 
be 18 51'. Index correction was + 3', and height of eye, 
15 feet. C. - W. was 4 ft 51 m 50* ; and the chr. correction 
by the rate card was 4*, slow. Patent log, 69.1. At 9 : 45 
by the watch, the sun's azimuth was again observed with 
pelorus, and found to be 137, compass bearing. It was 
intended to work a Sumner line from the altitude by Kelvin's 
table; and the pelorus observation was made because the 
sun's true azimuth always comes out as a by-product, when 
Kelvin's table is used, and so it is just as well to have an- 
other check on the deviation table. This is the peculiar 
advantage of Kelvin's table... Without any additional cal- 
culations, the compass is always checked up on the very 
course the ship is steering. This is just what the good 
navigator wants. 

The observations could not be worked up at once, be- 
cause the captain wished to see the result of the mate's 
bow-and-beam bearings. At 9 : 57 by the watch, Barnegat 
bore abeam, on the starboard hand, or 270 by compass, the 
yacht being still on the 180 compass course. Patent log 
now 72.5. 

E 



146 NAVIGATION 

Between the bow-and-beam bearings the run by log was 
72.5 67 = 5.5 miles. Therefore the yacht is now 5.5 
miles from Barnegat Light, and the compass bearing of the 
light is 270. The compass error being + 8, the true bear- 
ing of the light is 278 ; and the bearing of the yacht from 
the light is the former bearing reversed, or 278 180 = 98, 
true. From this comes an accurate and complete position 
of the yacht. Barnegat Light is in lat. 39 46' N. ; long. 74 6' 
W. The yacht, 5.5 miles away on the bearing 98, must, by 
traverse table, be in lat. 39 45' N. ; long. 73 59' W. 

At 10 A.M., the log was 73.4, course 188, true. 

Now the captain prepared to shape a new course to be 
followed from the Barnegat bow-and-beam bearing "fix" in 
the above lat. 39 45' N. ; long. 73 59' W., at 9 : 57. 

Allowing ten minutes to work up the new course, the 
captain plans to change course at 10 : 07. At that time 
the ship, on her course of 188, will be (at 15-knot speed) 
2'.5 S. and practically 0' W. of the Barnegat position. So 
the course will be changed when the yacht is in lat. 39 42' N. ; 
long. 73 59' W., at 10 : 07. The course and distance from 
there to the point 12 miles east of Hinchinbroke Rock are : 
distance, 945 miles ; course, 181, true, or 173 by compass. 

Therefore, by the table on page 52, the quartermaster gets 
the new course S4E. by compass, at 10 : 07. This corre- 
sponds to 174 by compass, or 182 true course; and at 
10 : 07, when the course was changed, the patent log read 
75.3. 

Now the Sumner line, from the observation at 9* 42 m 28* 
by the watch, was worked by Kelvin's table ; and the result 
was : 

Sumner point is in lat. 39 50' N. ; long. 73 56' W. ; bearing of 
Sumner line 237. 

It is necessary, as a check, to ascertain whether this Sum- 
ner line passes through the position obtained for the ship 
by the Barnegat bearings. Before doing this, the Sumner 
point must be shifted by the method of page 137, to allow for 



A NAVIGATOR'S DAY AT SEA 147 

the motion of the yacht between 9 : 42, when the sextant 
observation was made, and 9 : 57, when Barnegat bore 
abeam. The difference is 15 minutes, and in that time the 
ship moved south 3.4 miles by the patent log and an in- 
significant distance west. 

Therefore the corrected Sumner data are : 

Sumner point is in lat. 39 46'.6 N. ; long. 73 56' W. ; bearing of 
Sumner line 237. 

If everything fits, this Sumner line must pass through the 
Barnegat "fix" of the yacht in lat. 39 45' N. ; long. 73 59' 
W., because the yacht must have been somewhere on the 
line. 

The traverse table shows that the bearing of a line passing 
the Sumner point and the yacht's position is 235, differing 
only 2 from the Sumner line bearing ; so this check is satis- 
factory. But a better way to check this matter is to deter- 
mine the yacht's position from the intersection of two lines, 
one of which is the Sumner line, and the other the beam bear- 
ing of Barnegat Light. This can be done by the method of 
page 133. The data of the problem are : 

Sumner point : lat. 39 46'.6 N. 

long. 73 56' W. 
Line bears 237 
Barnegat Light : lat. 39 46' N. 

long. 74 6' W. 
Line bears 98 

We shall call Barnegat Light S r ; and then formula (3), 
page 136, gives, for the two bow bearings : 

At Sumner point, S, 237 - 266 = 29. 
At Barnegat, S', 98 - 266 = 168. 

For these two bearings, Table 14 gives the factor 0.74, and 
the yacht is placed 6 miles from Barnegat, on the 98 bear- 
ing. The bow-and-beam observations gave 5.5 miles, so 
the check by the Sumner line is excellent. 

It remains for the captain to utilize the azimuth observa- 



148 NAVIGATION 

tion made at 9 : 45. The bearing of the Sumner line was 
237, and therefore the sun's true azimuth was 147. The 
observed azimuth, by pelorus (p. 145), was 137. The com- 
pass error was therefore + 10. The variation being - 10, 
the deviation by formula (1), page 45, is D = 10 - ( - 10) = 
+ 20. 

On page 143 we found that the deviation table made this 
deviation + 18 ; so that the table appears to require a 
correction of +2. The captain decides not to correct 
the table for the present, unless later azimuth observations 
shall confirm it, especially as the sunrise observation showed 
the adjuster's results to be correct. Azimuth observa- 
tions made when the sun is high in the sky are not quite 
as reliable as sunrise ones. Moreover, the observation was 
made at 9 : 45, whereas the altitude observation, for which 
the true azimuth was calculated with Kelvin's table, was 
made at 9 : 42, so that the true azimuth must have been in 
error by the sun's azimuth change in three minutes. This 
could have been avoided by giving the mate orders to ob- 
serve the azimuth at about the same moment when the 
captain took the altitude. Or, the sun's azimuth change 
in three minutes might be taken from the azimuth table, and 
the computed true azimuth duly corrected. 

At 11 the log read 88.7, and the course was S.|E. by com- 
pass, or 182, true. 

At about 11 : 30, the weather showing signs of becoming 
thick, no preparations were made for a noon-sight by the 
method of page 86 ; and rather than take the risk of losing his 
noon observation altogether, the captain took an ex-me- 
ridian altitude at ll ft 42 TO 0* by his watch; log was 98.5; 
the sextant reading 26 55' ; index + 3' ; height of eye 15 
ft. ; C. W. was now 4* 51 m 42* ; and chronometer slow 4*. 

The observation was worked by Kelvin's table, and gave 
the Sumner point in lat. 39 20' N. ; long. 73 40' W. ; bearing 
of Sumner line 86. Figure 21 is a rough sketch of this Sumner 
line. It is very nearly horizontal ; had the observation been 



A NAVIGATOR'S DAY AT SEA 



149 



L 

Ship's 
Position 




39204 



made at noon precisely, it would have been perfectly hori- 
zontal. 

It would now have been possible to move up the Sumner 
line observed at 9 : 42, and obtain an intersection to fix the 
position of the yacht. 

But this did not seem 73J4o' 

necessary to the cap- 
tain, because of the 
beam bearing obtained 
at Barnegat at 9 : 57, 
which gave a good fix. 

And the present 
Sumner line being so 
nearly horizontal, it is 
not necessary to know 
the longitude very ac- 
curately to obtain an 
exact latitude. The 
longitude by D. R. is 
sufficient, and it is 73 58' W. The difference between 
this longitude and that of the Sumner point (73 40') is 
18' ; and the ship at L (fig. 21) bears 180 + 86 = 266 
from the Sumner point. Table 2 gives the dep. 14.0 for 
long. diff. 18', in lat. 39. And for course 266, dep. 14.0, 
we find in Table 1, lat. diff. I'.O, so the yacht's latitude is 1' 
less than that of the Sumner point, and is therefore 39 19'. 
This happens to be in exact accord with the D. R. latitude, 
which was also 39 19'. This was perfectly satisfactory, 
and the captain decided to carry this Sumner line forward 
for an intersection, in case he should obtain an observation 
in the afternoon. 

At 12, the patent log read 102.6, course S.fE., 182 true ; 
D. R. lat. 39 15' ; long. 73 58' ; distance to Watlings Island 
918 miles. 

Had the yacht been on a course other than almost due 
south, it would have been necessary to set the watch and the 



FIG. 21. Sumner Line from ex-Meridian 
Observation. 



150 NAVIGATION 

cabin clock to ship's apparent time. In fact, some naviga- 
tors set their watches to ship's apparent time before every 
observation (p. 94) : 

at 1, log read 117.7, misty, 
at 2, log read 133.0, misty, 
at 3, log read 149.0 misty, 
at 4, log read 163.8, clearing. 

At 4* I2 m 18 s by the watch, the weather having cleared, 
the altitude of the sun was found to be 4 38' ; index + 4' ; 
eye 15 ft. ; C. W. 4* 51 m 50* ; chronometer slow 4* ; log 
166.9. Sun's azimuth, observed by the mate at the same 
time, came out 224 by compass. 

This observation was worked for a Sumner line by the 
Kelvin table, and gave : 

Position of Sumner point lat. 38 6' N. ; long. 73 49' W. ; bearing 
of line 145 ; azimuth of sun 235. 

The Sumner line obtained at 11* 42 m 0' was brought up to 
the time of the present observation by D. R. (p. 137), giving : 

position of 11:42 Sumner 

point, after moving it, lat. 

38 12' N.; long. 73 43' W. ; 

bearing of the line 86. 

Both lines were then 

sketched, as shown in Fig. 

22. The point S is the 
FIG. 22. Rough Sketch of Sumner (moved) Sumner point from 
Line Intersection. ^ U:42 observation) S ' 

that from the 4 : 12 observation. The intersection point L is 
the position of the ship at 4 : 12, and it came out (p. 134) : 
lat. 38 11' N. ; long. 73 54' W. The position brought up 
by D. R. from 11 :42 was : lat. 38 11' ; long. 74 1' ; so that 
there has been an easterly set of the current, amounting to 
7' of longitude in 4| hours. The sun's true azimuth at 
4 : 12 was 235, from the Kelvin table ; and the pelorus 
observation gave 224. The compass error was therefore 




A NAVIGATOR'S DAY AT SEA 151 

+ 11. The variation being 10, the deviation must 
be D = 11 - ( - 10 =) + 21. The deviation table made 
this deviation + 18, so that table seems to require a correc- 
tion of +3. The pelorus observation of 9 : 45 gave a correc- 
tion of -f- 2 for the deviation table ; and as this is now 
apparently confirmed, the captain decides to examine the 
chart again, before finally shaping course for the night, to 
see if the yacht has not perhaps moved into a region where 
the variation is different from the Sandy Hook variation so 
far used. 

At 5 the log read 182.0, course was still 182 true. 

The captain now prepared to shape the course for the 
night, and to change his course, if necessary, at 6 : 00. His 
first step was to obtain the D. R. position at 6 : 00, starting 
from the observed position at 4 : 12. This gave position at 
6 : 00, by D. R. : lat. 37 41' ; long. 73 55'. The easterly 
current l of about 2' per hour set the yacht farther east about 
3' between 4 : 12 and 6 : 00. Therefore he took the D. R. 
position at 6 : 00 to be lat. 37 41' ; long. 73 52'. The posi- 
tion of the point of destination, 12 miles east of Watlings 
Island, is still : lat. 23 57' ; long. 74 15'. The true course 
and distance to that point from the yacht's 6 : 00 position is 
therefore, by traverse table : course 181| ; dist. 824 miles. 

A further examination of the track chart shows that the 
variation, which was 10 at Sandy Hook, is now 8. 
The compass error, from the last pelorus observation, 
was + 11. Consequently, by the pelorus observation, the 
compass course for the night should be 181| 11 = 170^, 
or S.fE. (see the Table on p. 52). Furthermore, the 
variation being now 8 and the error + 11 makes the 
deviation Z)=#-F= + ll-(-8) = + 19. The com- 
pass adjuster's deviation of + 18 is therefore vindicated, 
and the compass course S.fE. can be set for the night. 
At 6 the log read 197.2, course S.fE., or 182J true. 
1 Doubtless the Gulf Stream. 



152 NAVIGATION 

In conclusion, the captain of the Nav hopes he has been 
able to make his imagined proceedings clear enough to help 
the young navigator in planning his own first day's work at 
sea. May it be the first of many happy and successful days. 
And let him not forget, when attempting to verify the 
various calculations and problems of the Nav, that every 
observation in this book has been prepared by calculation, 
and none is the result of actual sextant observing. Should 
inconsistencies or errors be found by any young navigator, it 
is hoped that he will make them known so that they may be 
corrected, in case the Nav shall be required to make another 
voyage in a second edition. 



LIST OF TABLES 

1. Traverse Table; explained on pages 10 and 19; and its 

use in the Sumner method on pages 113, 135 154 

2. Conversion of longitude difference and departure ; ex- 

plained on page 16 168 

3. Number logarithms ; explained on page 23 178 

4. Trigonometric logarithms ; explained on page 31 196 

5. Meridional parts ; explained on page 35 241 

6. Sextant Correction Table ; explained on page 72 247 

7. Dip correction ; explained on page 73 247 

8. Conversion of hours and minutes into decimals of a day ; 

explained on page 80 248 

9. Conversion of degrees and minutes of longitude and hours 

and minutes of time 249 

10. Haversines ; explained on page 99 250 

11. Azimuth Table ; explained on page 113 284 

12. Auxiliary Azimuth Table; explained on page 115 290 

13. Kelvin's Sumner Line Table ; explained on page 126 292 

14. Sumner Intersection Table; explained on page 135 318 



PUBLISHERS' NOTE 

Table 3, Number Logarithms, has been reprinted from "The 
Macmillan Logarithmic and Trigonometric Tables," New York, 
1917. 



153 



154 



Table 1. Traverse Table 





1 


2 


i Pt. 3 


4 


5 


Pt. 6 


7 




(179, 181, 


(178, 182 


(177, 183, 


(176, 184 


(175, 185" 


(174, 186 


(173, 187, 


DlST 


359) 


358) 


357) 


356) 


355) 


354) 


353) 




Lat. 


Dep. 


Lat. 


Dep 


Lat. 


Dep. 


Lat. 


Dep 


Lat. 


Dep. 


Lat. 


Dep 


Lat. 


Dep. 


1 


1.0 


0.0 


1.0 


0.0 


1.0 


0.1 


1.0 


0.1 


1.0 


0.1 


1.0 


0.1 


1.0 


0.1 


2 


2.0 


0.0 


2.0 


0.1 


2.0 


0.1 


2.0 


0.1 


2.0 


0.2 


2.0 


0.2 


2.0 


0.2 


3 


3.0 


0.1 


3.0 


0.1 


3.0 


0.2 


3.0 


0.2 


3.0 


0.3 


3.0 


0.3 


3.0 


0.4 


4 


4.0 


0.1 


4.0 


0.1 


4.0 


0.2 


4.0 


0.3 


4.0 


0.3 


4.0 


0.4 


4.0 


0.5 


5 


5.0 


0.1 


5.0 


0.2 


5.0 


0.3 


5.0 


0.3 


5.0 


0.4 


5.0 


0.5 


5.0 


0.6 


6 


6.0 


0.1 


6.0 


0.2 


6.0 


0.3 


6.0 


0.4 


6.0 


0.5 


6.0 


0.6 


6.0 


0.7 


7 


7.0 


0.1 


7.0 


0.2 


7.0 


0.4 


7.0 


0.5 


7.0 


0.6 


7.0 


0.7 


6.9 


0.9 


8 


8.0 


0.1 


8.0 


0.3 


8.0 


0.4 


8.0 


0.6 


8.0 


0.7 


8.0 


0.8 


7.9 


1.0 


9 


9.0 


0.2 


9.0 


0.3 


9.0 


0.5 


9.0 


0.6 


9.0 


0.8 


9.0 


0.9 


8.9 


1.1 


10 


10.0 


0.2 


10.0 


0.3 


10.0 


0.5 


10.0 


0.7 


10.0 


0.9 


9.9 


1.0 


9.9 


1.2 


11 


11.0 


0.2 


11.0 


0.4 


11.0 


0.6 


11.0 


0.8 


11.0 


1.0 


10.9 


1.1 


10.9 


1.3 


12 


12.0 


0.2 


12.0 


0.4 


12.0 


0.6 


12.0 


0.8 


12.0 


1.0 


11.9 


1.3 


11.9 


1.5 


13 


13.0 


0.2 


13.0 


0.5 


13.0 


0.7 


13.0 


0.9 


13.0 


1.1 


12.9 


1.4 


12.9 


1.6 


14 


14.0 


0.2 


14.0 


0.5 


14.0 


0.7 


14.0 


1.0 


13.9 


1.2 


13.9 


1.5 


13.9 


1.7 


15 


15.0 


0.3 


15.0 


0.5 


15.0 


0.8 


15.0 


1.0 


14.9 


1.3 


14.9 


1.6 


14.9 


1.8 


16 


16.0 


0.3 


16.0 


0.6 


16.0 


0.8 


16.0 


1.1 


15.9 


1.4 


15.9 


1.7 


15.9 


1.9 


17 


17.0 


0.3 


17.0 


0.6 


17.0 


0.9 


17.0 


1.2 


16.9 


1.5 


16.9 


1.8 


16.9 


2.1 


18 


18.0 


0.3 


18.0 


0.6 


18.0 


0.9 


18.0 


1.3 


17.9 


1.6 


17.9 


1.9 


17.9 


2.2 


19 


19.0 


0.3 


19.0 


0.7 


19.0 


1.0 


19.0 


1.3 


18.9 


1.7 


18.9 


2.0 


18.9 


2.3 


20 


20.0 


0.3 


20.0 


0.7 


20.0 


1.0 


20.0 


1.4 


19.9 


1.7 


19.9 


2.1 


19.9 


2.4 


21 


21.0 


0.* 


21.0 


0.7 


21.0 


1.1 


20.9 


1.5 


20.9 


1.8 


20.9 


2.2 


20.8 


2.6 


22 


22.0 


0.4 


22.0 


0.8 


22.0 


1.2 


21.9 


1.5 


21.9 


1.9 


21.9 


2.3 


21.8 


2.7 


23 


23.0 


0.4 


23.0 


0.8 


23.0 


1.2 


22.9 


1.6 


22.9 


2.0 


22.9 


2.4 


22.8 


2.8 


24 


24.0 


0.4 


24.0 


0.8 


24.0 


1.3 


23.9 


1.7 


23.9 


2.1 


23.9 


2.5 


23.8 


2.9 


25 


25.0 


0.4 


25.0 


0.9 


25.0 


1.3 


24.9 


1.7 


24.9 


2.2 


24.9 


2.6 


24.8 


3.0 


26 


26.0 


0.5 


26.0 


0.9 


26.0 


1.4 


25.9 


1.8 


25.9 


2.3 


25.9 


2.7 


25.8 


3.2 


27 


27.0 


0.5 


27.0 


0.9 


27.0 


1.4 


26.9 


1.9 


26.9 


2.4 


26.9 


2.8 


26.8 


3.3 


28 


28.0 


0.5 


28.0 


1.0 


28.0 


1.5 


27.9 


2.0 


27.9 


2.4 


27.8 


2.9 


27.8 


3.4 


29 


29.0 


0.5 


29.0 


1.0 


29.0 


1.5 


28.9 


2.0 


28.9 


2.5 


28.8 


3.0 


28.8 


3.5 


30 


30.0 


0.5 


30.0 


1.0 


30.0 


1.6 


29.9 


2.1 


29.9 


2.6 


29.8 


3.1 


29.8 


3.7 


31 


31.0 


0.5 


31.0 


1.1 


31.0 


1.6 


30.9 


2.2 


30.9 


2.7 


30.8 


3.2 


30.8 


3.8 


32 


32.0 


0.6 


32.0 


1.1 


32.0 


1.7 


31.9 


2.2 


31.9 


2.8 


31.8 


3.3 


31.8 


3.9 


33 


33.0 


0.6 


33.0 


1.2 


33.0 


1.7 


32.9 


2.3 


32.9 


2.9 


32.8 


3.4 


32.8 


4.0 


34 


34.0 


0.6 


34.0 


1.2 


34.0 


1.8 


33.9 


2.4 


33.9 


3.0 


33.8 


3.6 


33.7 


4.1 


35 


35.0 


0.6 


35.0 


1.2 


35.0 


1.8 


34.9 


2.4 


34.9 


3.1 


34.8 


3.7 


34.7 


4.3 


36 


36.0 


0.6 


36.0 


1.3 


36.0 


1.9 


35.9 


2.5 


35.9 


3.1 


35.8 


3.8 


35.7 


4.4 


37 


37.0 


0.6 


37.0 


1.3 


36.9 


1.9 


36.9 


2.6 


36.9 


3.2 


36.8 


3.9 


36.7 


4.5 


38 


38.0 


0.7 


38.0 


1.3 


37.9 


2.0 


37.9 


2.7 


37.9 


3.3 


37.8 


4.0 


37.7 


4.6 


39 


39.0 


0.7 


39.0 


1.4 


38.9 


2.0 


38.9 


2.7 


38.9 


3.4 


38.8 


4.1 


38.7 


4.8 


40 


40.0 


0.7 


40.0 


1.4 


39.9 


2.1 


39.9 


2.8 


39.8 


3.5 


39.8 


4.2 


39.7 


4.9 


41 


41.0 


0.7 


41.0 


1.4 


40.9 


2.1 


40.9 


2.9 


40.8 


3.6 


40.8 


4.3 


40.7 


5.0 


42 


42.0 


0.7 


42.0 


1.5 


41.9 


2.2 


41.9 


2.9 


41.8 


3.7 


41.8 


4.4 


41.7 


5.1 


43 


43.0 


0.8 


43.0 


1.5 


42.9 


2.3 


42.9 


3.0 


42.8 


3.7 


42.8 


4.5 


42.7 


5.2 


44 


44.0 


0.8 


44.0 


1.5 


43.9 


2.3 


43.9 


3.1 


43.8 


3.8 


43.8 


4.6 


43.7 


5.4 


45 


45.0 


0.8 


45.0 


1.6 


44.9 


2.4 


44.9 


3.1 


44.8 


3.9 


44.8 


4.7 


44.7 


5.5 


46 


46.0 


0.8 


46.0 


1.6 


45.9 


2.4 


45.9 


3.2 


45.8 


4.0 


45.7 


4.8 


45.7 


5.6 


47 


47.0 


0.8 


47.0 


1.6 


46.9 


2.5 


46.9 


3.3 


46.8 


4.1 


46.7 


4.9 


46.6 


5.7 


48 


48.0 


0.8 


48.0 


1.7 


47.9 


2.5 


47.9 


3.3 


47.8 


4.2 


47.7 


5.0 


47.6 


5.8 


49 


49.0 


0.9 


49.0 


1.7 


48.9 


2.6 


48.9 


3.4 


48.8 


4.3 


48.7 


5.1 


48.6 


6.0 


50 


50.0 


0.9 


50.0 


1.7 


49.9 


2.6 


49.9 


3.5 


49.8 


4.4 


49.7 


5.2 


49.6 


6.1 


100 


100.0 


1.7 


99.9 


3.5 


99.9 


5.2 


99.8 


7.0 


99.6 


8.7 


99.5 


10.5 


99.3 


12.2 


200 


200.0 


3.5 


199.9 


7.0 


199.7 


10.5 


199.5 


14.0 


199.2 


17.4 


198.9 


20.9 


198.5 


24.4 


300 


300.0 


5.2 


299.8 


10.5 


299.6 


15.7 


299.3 


20.9 


298.9 


26.1 


298.4 


31.4 


297.8 


36.6 


400 


399.9 


7.0 


399.8 


13.9 


399.4 


20.9 


399.0 


27.9 


398.5 


34.9 


397.8 


41.8 


397.0 


48.7 


500 


499.9 


8.8 


499.7 


17.4 


499.3 


26.2 


498.8 


34.8 


498.1 


43.6 


497.3 


52.3 


496.3 


61.0 




Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 




(91, 269, 


(92, 268, 


(93, 267, 


(94, 266, 


(95, 265, 


(96, 264, 


(97, 263, 




271) 


272) 


273) 


274) 


275) 


276) 


277) 




89 


88 


7fPt.87 


86 


85 


7|Pt.84 


83 



Table 1. Traverse Table 



155 





1 


2 


| Pt. 3 


4 


5 


Pt. 6 


7 




(179, 181 


(178, 182, 


(177, 183, 


(176, 184, 


(175, 185, 


(174, 186, 


(173, 187, 


DlBT 


359) 


358) 


357) 


356) 


355) 


354) 


353) 




Lat. 


Dep 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


51 


51.0 


0.9 


51.0 


1.8 


50.9 


2.7 


50.9 


3.6 


50.8 


4.4 


50.7 


5.3 


50.6 


6.2 


52 


52.0 


0.9 


52.0 


1.8 


51.9 


2.7 


51.9 


3.6 


51.8 


4.5 


51.7 


5.4 


51.6 


6.3 


53 


53.0 


0.9 


53.0 


1.8 


52.9 


2.8 


52.9 


3.7 


52.8 


4.6 


52.7 


5.5 


52.6 


6.5 


54 


54.0 


0.9 


54.0 


1.9 


53.9 


2.8 


53.9 


3.8 


53.8 


4.7 


53.7 


5.6 


53.6 


6.6 


55 


55.0 


1.0 


55.0 


1.9 


54.9 


2.9 


54.9 


3.8 


54.8 


4.8 


54.7 


5.7 


54.6 


6.7 


56 


56.0 


1.0 


56.0 


2.0 


55.9 


2.9 


55.9 


3.9 


55.8 


4.9 


55.7 


5.9 


55.6 


6.8 


57 


57.0 


1.0 


57.0 


2.0 


56.9 


3.0 


56.9 


4.0 


56.8 


5.0 


56.7 


6.0 


56.6 


6.9 


58 


58.0 


1.0 


58.0 


2.0 


57.9 


3.0 


57.9 


4.0 


57.8 


5.1 


57.7 


6.1 


57.6 


7.1 


59 


59.0 


1.0 


59.0 


2.1 


58.9 


3.1 


58.9 


4.1 


58.8 


5.1 


58.7 


6.2 


58.6 


7.2 


60 


60.0 


1.0 


60.0 


2.1 


59.9 


3.1 


59.9 


4.2 


59.8 


5.2 


59.7 


6.3 


59.6 


7.3 


61 


61.0 


1.1 


61.0 


2.1 


60.9 


3.2 


60.9 


4.3 


60.8 


5.3 


60.7 


6.4 


60.5 


7.4 


62 


62.0 


1.1 


62.0 


2.2 


61.9 


3.2 


61.8 


4.3 


61.8 


5.4 


61.7 


6.5 


61.5 


7.6 


63 


63.0 


1.1 


63.0 


2.2 


62.9 


3.3 


62.8 


4.4 


62.8 


5.5 


62.7 


6.6 


62.5 


7.7 


64 


64.0 


1.1 


64.0 


2.2 


63.9 


3.3 


63.8 


4.5 


63.8 


5.6 


63.6 


6.7 


63.5 


7.8 


65 


65.0 


1.1 


65.0 


2.3 


64.9 


3.4 


64.8 


4.5 


64.8 


5.7 


64.6 


6.8 


64.5 


7.9 


66 


66.0 


1.2 


66.0 


2.3 


65.9 


3.5 


65.8 


4.6 


65.7 


5.8 


65.6 


6.9 


65.5 


8rO 


67 


67.0 


1.2 


67.0 


2.3 


66.9 


3.5 


66.8 


4.7 


66.7 


5.8 


66.6 


7.0 


66.5 


8.2 


68 


68.0 


1.2 


68.0 


2.4 


67.9 


3.6 


67.8 


4.7 


67.7 


5.9 


67.6 


7.1 


67.5 


8.3 


69 


69.0 


1.2 


69.0 


2.4 


68.9 


3.6 


68.8 


4.8 


68.7 


6.0 


68.6 


7.2 


68.5 


8.4 


70 


70.0 


1.2 


70.0 


2.4 


69.9 


3.7 


69.8 


4.9 


69.7 


6.1 


69.6 


7.3 


69.5 


8.5 


71 


71.0 


1.2 


71.0 


2.5 


70.9 


3.7 


70.8 


5.0 


70.7 


6.2 


70.6 


7.4 


70.5 


8.7 


72 


72.0 


1.3 


72.0 


2.5 


71.9 


3.8 


71.8 


5.0 


71.7 


6.3 


71.6 


7 5 


71.5 


8.8 


73 


73.0 


1.3 


73.0 


2.5 


72.9 


3.8 


72.8 


5.1 


72.7 


6.4 


72.6 


7.6 


72.5 


8.9 


74 


74.0 


1.3 


74.0 


2.6 


73.9 


3.9 


73.8 


5.2 


73.7 


6.4 


73.6 


7.7 


73.4 


9.0 


75 


75.0 


1.3 


75.0 


2.6 


74.9 


3.9 


74.8 


5.2 


74.7 


6.5 


74.6 


7.8 


74.4 


9.1 


76 


76.0 


1.3 


76.0 


2.7 


75.9 


4.0 


75.8 


5.3 


75.7 


6.6 


75.6 


7.9 


75.4 


9.3 


77 


77.0 


1.3 


77.0 


2.7 


76.9 


4.0 


76.8 


5.4 


76.7 


6.7 


76.6 


8.0 


76.4 


9.4 


78 


78.0 


1.4 


78.0 


2.7 


77.9 


4.1 


77.8 


5.4 


77.7 


6.8 


77.6 


8.2 


77.4 


9.5 


79 


79.0 


1.4 


79.0 


2.8 


78.9 


4.1 


78.8 


5.5 


78.7 


6.9 


78.6 


8.3 


78.4 


9.6 


80 


80.0 


1.4 


80.0 


2.8 


79.9 


4.2 


79.8 


5.6 


79.7 


7.0 


79.6 


8.4 


79.4 


9.7 


81 


81.0 


1.4 


81.0 


2.8 


80.9 


4.2 


80.8 


5.7 


80.7 


7.1 


80.6 


8.5 


80.4 


9.9 


82 


82.0 


1.4 


82.0 


2.9 


81.9 


4.3 


81.8 


5.7 


81.7 


7.1 


81.6 


8.6 


81.4 


10.0 


83 


83.0 


1.4 


82.9 


2.9 


82.9 


4.3 


82.8 


5.8 


82.7 


7.2 


82.5 


8.7 


82.4 


10.1 


84 


84.0 


1.5 


83.9 


2.9 


83.9 


4.4 


83.8 


5.9 


83.7 


7.3 


83.5 


8.8 


83.4 


10.2 


85 


85.0 


1.5 


84.9 


3.0 


84.9 


4.4 


84.8 


5.9 


84.7 


7.4 


84.5 


8.9 


84.4 


10.4 


86 


86.0 


1.5 


85.9 


3.0 


85.9 


4.5 


85.8 


6.0 


85.7 


7.5 


85.5 


9.0 


85.4 


10.5 


87 


87.0 


1.5 


86.9 


3.0 


86.9 


4.6 


86.8 


6.1 


86.7 


7.6 


86.5 


9.1 


86.4 


10.6 


88 


88.0 


1.5 


87.9 


3.1 


87.9 


4.6 


87.8 


6.1 


87.7 


7.7 


87.5 


9.2 


87.3 


10.7 


89 


89.0 


1.6 


88.9 


3.1 


88.9 


4.7 


88.8 


6.2 


88.7 


7.8 


88.5 


9.3 


88.3 


10.8 


90 


90.0 


1.6 


89.9 


3.1 


89.9 


4.7 


89.8 


6.3 


89.7 


7.8 


89.5 


9.4 


89.3 


11.0 


91 


91.0 


1.6 


90.9 


3.2 


90.9 


4.8 


90.8 


6.3 


90.7 


7.9 


90.5 


9.5 


90.3 


11.1 


92 


92.0 


1.6 


91.9 


3.2 


91.9 


4.8 


91.8 


6.4 


91.6 


8.0 


91.5 


9.6 


91.3 


11.2 


93 


93.0 


1.6 


92.9 


3.2 


92.9 


4.9 


92.8 


6.5 


92.6 


8.1 


92.5 


9.7 


92.3 


11.3 


94 


94.0 


1.6 


93.9 


3.3 


93.9 


4.9 


93.8 


6.6 


93.6 


8.2 


93.5 


9.8 


93.3 


11.5 


95 


95.0 


1.7 


94.9 


3.3 


94.9 


5.0 


94.8 


6.6 


94.6 


8.3 


94.5 


9.9 


94.3 


11.6 


96 


96.0 


1.7 


95.9 


3.4 


95.9 


5.0 


95.8 


6.7 


95.6 


8.4 


95.5 


10.0 


95.3 


11.7 


97 


97.0 


1.7 


96.9 


3.4 


96.9 


5.1 


96.8 


6.8 


96.6 


8.5 


96.5 


10.1 


96.3 


11.8 


98 


98.0 


1.7 


97.9 


3.4 


97.9 


5.1 


97.8 


6.8 


97.6 


8.5 


97.5 


10.2 


97.3 


11.9 


99 


99.0 


1.7 


98.9 


3.5 


98.9 


5.2 


98.8 


6.9 


98.6 


8.6 


98.5 


10.3 


98.3 


12.1 


100 


100.0 


1.7 


99.9 


3.5 


99.9 


5.2 


99.8 


7.0 


99.6 


8.7 


99.5 


10.5 


99.3 


12.2 


600 


599.9 


10.5 


599.6 


20.9 


599.2 


31.4 


598.6 


41.9 


597.7 


52.3 


596.7 


62.7 


595.5 


73.1 


700 


699.8 


12.2 


699.5 


24.4 


699.0 


36.6 


698.2 


48.8 


697.2 


61.0 


696.1 


73.2 


694.9 


85.3 


800 


799.8 


14.0 


799.5 


27.9 


798.9 


41.9 


798.0 


55.8 


796.9 


69.7 


795.6 


83.6 


794.1 


97.5 


900 


899.7 


15.7 


899.3 


31.4 


898.6 


47.1 


897.6 


62.8 


896.4 


78.4 


895.0 


94.1 


893.3 


109.6 




Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 




(91, 269, 


(92, 268 


(93, 267, 


(94, 266, 


(95, 265, 


(96, 264, 


(97, 263. 




271) 


272) 


273) 


274) 


275) 


276) 


277) 




89 


88 


71 Pt. 87 


86 


85 


7i Pt. 84 


83 



156 



Table 1. Traverse Table 





f Pt. 8 


9 


10 


1 Pt. 11 


12 


13 


1 \ Pt, 14 




(172, 188, 


(171, 189, 


(170, 190, 


(169, 191, 


(168, 192, 


(167, 193, 


(166, 194, 


DlST. 


352) 


351) 


350) 


349) 


348) 


347) 


346) 




Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


1 


1.0 


0.1 


1.0 


0.2 


1.0 


0.2 


1.0 


0.2 


1.0 


0.2 


1.0 


0.2 


1.0 


0.2 


2 


2.0 


0.3 


2.0 


0.3 


2.0 


0.3 


2.0 


0.4 


2.0 


0.4 


1.9 


0.4 


1.9 


0.5 


3 


3.0 


0.4 


3.0 


0.5 


3.0 


0.5 


2.9 


0.6 


2.9 


0.6 


2.9 


0.7 


2.9 


0.7 


4 


4.0 


0.6 


4.0 


0.6 


3.9 


0.7 


3.9 


0.8 


3.9 


0.8 


3.9 


0.9 


3.9 


1.0 


5 


5.0 


0.7 


4.9 


0.8 


4.9 


0.9 


4.9 


1.0 


4.9 


1.0 


4.9 


1.1 


4.9 


1.2 


6 


5.9 


0.8 


5.9 


0.9 


5.9 


1.0 


5.9 


1.1 


5.9 


1.2 


5.8 


1.3 


5.8 


1.5 


7 


6.9 


1.0 


6.9 


1.1 


6.9 


1.2 


6.9 


1.3 


6.8 


1.5 


6.8 


1.6 


6.8 


1.7 


8 


7.9 


1.1 


7.9 


1.3 


7.9 


1.4 


7.9 


1.5 


7.8 


1.7 


7.8 


1.8 


7.8 


1.9 


9 


8.9 


1.3 


8.9 


1.4 


8.9 


1.6 


8.8 


1.7 


8.8 


1.9 


8.8 


2.0 


8.7 


2.2 


10 


9.9 


1.4 


9.9 


1.6 


9.8 


1.7 


9.8 


1.9 


9.8 


2.1 


9.7 


2.2 


9.7 


2.4 


11 


10.9 


1.5 


10.9 


1.7 


10.8 


1.9 


10.8 


2.1 


10.8 


2.3 


10.7 


2.5 


10.7 


2.7 


12 


11.9 


1.7 


11.9 


1.9 


11.8 


2.1 


11.8 


2.3 


11.7 


2.5 


11.7 


2.7 


11.6 


2.9 


13 


12.9 


1.8 


12.8 


2.0 


12.8 


2.3 


12.8 


2.5 


12.7 


2.7 


12.7 


2.9 


12.6 


3.1 


14 


13.9 


1.9 


13.8 


2.2 


13.8 


2.4 


13.7 


2.7 


13.7 


2.9 


13.6 


3.1 


13.6 


3.4 


15 


14.9 


2.1 


14.8 


2.3 


14.8 


2.6 


14.7 


2.9 


14.7 


3.1 


14.6 


3.4 


14.6 


3.6 


16 


15.8 


2.2 


15.8 


2.5 


15.8 


2".8 


15.7 


3.1 


15.7 


3.3 


15.6 


3.6 


15.5 


3.9 


17 


16.8 


2.4 


16.8 


2.7 


16.7 


3.0 


16.7 


3.2 


16.6 


3.5 


16.6 


3.8 


16.5 


4.1 


18 


17.8 


2.5 


17.8 


2.9 


17.7 


3.1 


17.7 


3.4 


17.6 


3.7 


17.5 


4.0 


17.5 


4.4 


19 


18.8 


2.6 


18.8 


3.0 


18.7 


3.3 


18.7 


3.6 


18.6 


4.0 


18.5 


4.3 


18.4 


4.6 


20 


19.8 


2.8 


19.8 


3.1 


19.7 


3.5 


19.6 


3.8 


19.6 


4.2 


19.5 


4.5 


19.4 


4.8 


21 


20.8 


2.9 


20.7 


3.3 


20.7 


3.6 


20.6 


4.0 


20.5 


4.4 


20.5 


4.7 


20.4 


5.1 


22 


21.8 


3.1 


21.7 


3.4 


21.7 


3.8 


21.6 


4.2 


21.5 


4.6 


21.4 


4.9 


21.3 


5.3 


23 


22.8 


3.2 


22.7 


3.6 


22.7 


4.0 


22.6 


4.4 


22.5 


4.8 


22.4 


5.2 


22.3 


5.6 


24 


23.8 


3.3 


23.7 


3.8 


23.6 


4.2 


23.6 


4.6 


23.5 


5.0 


23.4 


5.4 


23.3 


5.8 


25 


24.8 


3.5 


24.7 


3.9 


24.6 


4.3 


24.5 


4.8 


24.5 


5.2 


24.4 


5.6 


24.3 


6.0 


26 


25.7 


3.6 


25.7 


4.1 


25.6 


4.5 


25.5 


5.0 


25.4 


5.4 


25.3 


5.8 


25.2 


6.3 


27 


26.7 


3.8 


26.7 


4.2 


26.6 


4.7 


26.5 


5.2 


26.4 


5.6 


26.3 


6.1 


26.2 


6.5 


28 


27.7 


3.9 


27.7 


4.4 


27.6 


4.9 


27.5 


5.3 


27.4 


5.8 


27.3 


6.3 


27.2 


6.8 


29 


28.7 


4.0 


28.6 


4.5 


28.6 


5.0 


28.5 


5.5 


28.4 


6.0 


28.3 


6.5 


28.1 


7.0 


30 


29.7 


4.2 


29.6 


4.7 


29.5 


5.2 


29.4 


5.7 


29.3 


6.2 


29.2 


6.7 


29.1 


7.3 


31 


30.7 


4.3 


30.6 


4.8 


30.5 


5.4 


30.4 


5.9 


30.3 


6.4 


30.2 


7.0 


30.1 


7.5 


32 


31.7 


4.5 


31.6 


5.0 


31.5 


5.6 


31.4 


6.1 


31.3 


6.7 


31.2 


7.2 


31.0 


7.7 


33 


32.7 


4.6 


32.6 


5.2 


32.5 


5.7 


32.4 


6.3 


32.3 


6.9 


32.2 


7.4 


32.0 


8.0 


34 


33.7 


4.7 


33.6 


5.3 


33.5 


5.9 


33.4 


6.5 


33.3 


7.1 


33.1 


7.6 


33.0 


8.2 


35 


34.7 


4.9 


34.6 


5.5 


34.5 


6.1 


34.4 


6.7 


34.2 


7.3 


34.1 


7.9 


34.0 


8.5 


36 


35.6 


5.0 


35.6 


5.6 


35.5 


6.3 


35.3 


6.9 


35.2 


7.5 


35.1 


8.1 


34.9 


8.7 


37 


36.6 


5.1 


36.5 


5.8 


36.4 


6.4 


36.3 


7.1 


36.2 


7.7 


36.1 


8.3 


35.9 


9.0 


38 


37.6 


5.3 


37.5 


5.9 


37.4 


6.6 


37.3 


7.3 


37.2 


7.9 


37.0 


8.5 


36.9 


9.2 


39 


38.6 


5.4 


38.5 


6.1 


38.4 


6.8 


38.3 


7.4 


38.1 


8.1 


38.0 


8.8 


37.8 


9.4 


40 


39.6 


6.6 


39.5 


6.3 


39.4 


6.9 


39.3 


7.6 


39.1 


8.3 


39.0 


9.0 


38.8 


9.7 


41 


40.6 


5.7 


40.5 


6.4 


40.4 


7.1 


40.2 


7.8 


40.1 


8.5 


39.9 


9.2 


39.8 


9.9 


42 


41.6 


5.8 


41.5 


6.6 


41.4 


7.3 


41.2 


8.0 


41.1 


8.7 


40.9 


9.4 


40.8 


10.2 


43 


42.6 


6.0 


42.5 


6.7 


42.3 


7.5 


42.2 


8.2 


42.1 


8.9 


41.9 


9.7 


41.7 


10.4 


44 


43.6 


6.1 


43.5 


6.9 


43.3 


7.6 


43.2 


8.4 


43.0 


9.1 


42.9 


9.9 


42.7 


10.6 


45 


44.6 


6.3 


44.4 


7.0 


44.3 


7.8 


44.2 


8.6 


44.0 


9.4 


43.8 


10.1 


43.7 


10.9 


46 


45.6 


6.4 


45.4 


7.2 


45.3 


8.0 


45.2 


8.8 


45.0 


9.6 


44.8 


10.3 


44.6 


11.1 


47 


46.5 


6.5 


46.4 


7.4 


46.3 


8.2 


46.1 


9.0 


46.0 


9.8 


45.8 


10.6 


45.6 


11.4 


48 


47.5 


6.7 


47.4 


7.5 


47.3 


8.3 


47.1 


9.2 


47.0 


10.0 


46.8 


10.8 


46.6 


11.6 


49 


48.5 


6.8 


48.4 


7.7 


48.3 


8.5 


48.1 


9.3 


47.9 


10.2 


47.7 


11.0 


47.5 


11.9 


50 


49.5 


7.0 


49.4 


7.8 


49.2 


8.7 


49.1 


9.5 


48.9 


10.4 


48.7 


11.2 


48.5 


12.1 


100 


99.0 


13.9 


98.8 


15.6 


98.5 


17.4 


98.2 


19.1 


97.8 


20.8 


97.4 


22.5 


97.0 


24.2 


200 


198.1 


27.8 


197.5 


31.3 


197.0 


34.7 


196.3 


38.2 


195.6 


41.6 


194.9 


45.0 


194.1 


48.4 


300 


297.1 


41.8 


296.3 


46.9 


295.4 


52.1 


294.5 


57.2 


293.4 


62.4 


292.3 


67.5 


291.1 


72.6 


400 


396.1 


55.7 


395.1 


62.6 


393.9 


69.5 


392.6 


76.3 


391.3 


83.1 


389.8 


90.0 


388.1 


96.7 


500 


495.1 


69.6 


493.8 


78.2 


492.4 


86.8 


490.8 


95.4 


489.1 


104.0 


487.2 


112.4 


485.1 


121.0 




Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 




(98, 262, 


(99, 261, 


(100, 260, 


(101, 259, 


(102, 258, 


(103, 257, 


(104, 256, 




278) 


279) 


280) 


281) 


282) 


283) 


284) 




7J Pt. 82 


81 


80 


7 Pt. 79 


78 


77 


6 f Pt. 76 



The 1-Pt. or 11 Courses are : N. by E., N. by W., S. by E., S. by W. 



Table 1. Traverse Table 



157 





f Pt. 8 


9 


10 


1 Pt. 11 


12 


13 


HPt. 14 




(172, 188, 


(171, 189, 


(170, 190, 


(169, 191, 


(168, 192, 


(167, 193, 


(166, 194, 


DlST. 


352) 


351) 


350) 


349) 


348) 


347) 


346) 




Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


51 


50.5 


7.1 


50.4 


8.0 


50.2 


8.9 


50.1 


9.7 


49.9 


10.6 


49.7 


11.5 


49.5 


12.3 


52 


51.5 


7.2 


51.4 


8.1 


51.2 


9.0 


51.0 


9.9 


50.9 


10.8 


50.7 


11.7 


50.5 


12.6 


53 


52.5 


7.4 


52.3 


8.3 


52.2 


9.2 


52.0 


10.1 


51.8 


11.0 


51.6 


11.9 


51.4 


12.8 


54 


53.5 


7.5 


53.3 


8.4 


53.2 


9.4 


53.0 


10.3 


52.8 


11.2 


52.6 


12.1 


52.4 


13.1 


55 


54.5 


7.7 


54.3 


8.6 


54.2 


9.6 


54.0 


10.5 


53.8 


11.4 


53.6 


12.4 


53.4 


13.3 


56 


55.5 


7.8 


55.3 


8.8 


55.1 


9.7 


55.0 


10.7 


54.8 


11.6 


54.6 


12.6 


54.3 


13.5 


57 


56.4 


7.9 


56.3 


8.9 


56.1 


9.9 


56.0 


10.9 


55.8 


11.9 


55.5 


12.8 


55.3 


13.8 


58 


57.4 


8.1 


57.3 


9.1 


57.1 


10.1 


56.9 


11.1 


56.7 


12.1 


56.5 


13.0 


56.3 


14.0 


59 


58.4 


8.2 


58.3 


9.2 


58.1 


10.2 


57.9 


11.3 


57.7 


12.3 


57.5 


13.3 


57.2 


14.3 


60 


59.4 


8.4 


59.3 


9.4 


59.1 


10.4 


58.9 


11.4 


58.7 


12.5 


58.5 


13.5 


58.2 


14.5 


61 


60.4 


8.5 


60.2 


9.5 


60.1 


10.6 


59.9 


11.6 


59.7 


12.7 


59.4 


13.7 


59.2 


14.8 


62 


61.4 


8.6 


61.2 


9.7 


61.1 


10.8 


60.9 


11.8 


60.6 


12.9 


60.4 


13.9 


60.2 


15.0 


63 


62.4 


8.8 


62.2 


9.9 


62.0 


10.9 


61.8 


12.0 


61.6 


13.1 


61.4 


14.2 


61.1 


15.2 


64 


63.4 


8.9 


63.2 


10.0 


63.0 


11.1 


62.8 


12.2 


62.6 


13.3 


62.4 


14.4 


62.1 


15.5 


65 


64.4 


9.0 


64.2 


10.2 


64.0 


11.3 


63.8 


12.4 


63.6 


13.5 


63.3 


14.6 


63.1 


15.7 


66 


65.4 


9.2 


65.2 


10.3 


65.0 


11.5 


64.8 


12.6 


64.6 


13.7 


64.3 


14.8 


64.0 


16.0 


67 


66.3 


9 3 


66.2 


10.5 


66.0 


11.6 


65.8 


12.8 


65.5 


13.9 


65.3 


15.1 


65.0 


16.2 


68 


67.3 


9.5 


67.2 


10.6 


67.0 


11.8 


66.8 


13.0 


66.5 


14.1 


66.3 


15.3 


66.0 


16.5 


69 


68.3 


9.6 


68.2 


10.8 


68.0 


12.0 


67.7 


13.2 


67.5 


14.3 


67.2 


15.5 


67.0 


16.7 


70 


69.3 


9.7 


69.1 


11.0 


68.9 


12.2 


68.7 


13.4 


68.5 


14.6 


68.2 


15.7 


67.9 


16.9 


71 


70.3 


9.9 


70.1 


11.1 


69.9 


12.3 


69.7 


13.5 


69.4 


14.8 


69.2 


16.0 


68.9 


17.2 


72 


71.3 


10.0 


71.1 


11.3 


70.9 


12.5 


70.7 


13.7 


70.4 


15.0 


70.2 


16.2 


69.9 


17.4 


73 


72.3 


10.2 


72.1 


11.4 


71.9 


12.7 


71.7 


13.9 


71.4 


15.2 


71.1 


16.4 


70.8 


17.7 


74 


73.3 


10.3 


73.1 


11.6 


72.9 


12.8 


72.6 


14.1 


72.4 


15.4 


72.1 


16.6 


71.8 


17.9 


75 


74.3 


10.4 


74.1 


11.7 


73.9 


13.0 


73.6 


14.3 


73.4 


15.6 


73.1 


16.9 


72.8 


18.1 


76 


75.3 


10.6 


75.1 


11.9 


74.8 


13.2 


74.6 


14.5 


74.3 


15.8 


74.1 


17.1 


73.7 


18.4 


77 


76.3 


10.7 


76.1 


12.0 


75.8 


13.4 


75.6 


14.7 


75.3 


16.0 


75.0 


17.3 


74.7 


18.6 


78 


77.2 


10.9 


77.0 


12.2 


76.8 


13.5 


76.6 


14.9 


76.3 


16.2 


76.0 


17.5 


75.7 


18.9 


79 


78.2 


11.0 


78.0 


12.4 


77.8 


13.7 


77.5 


15.1 


77.3 


16.4 


77.0 


17.8 


76.7 


19.1 


80 


79.2 


11.1 


79.0 


12.5 


78.8 


13.9 


78.5 


15.3 


78.3 


16.6 


77.9 


18.0 


77.6 


19.4 


81 


80.2 


11.3 


80.0 


12.7 


79.8 


14.1 


79.5 


15.5 


79.2 


16.8 


78.9 


18.2 


78.6 


19.6 


82 


81.2 


11.4 


81.0 


12.8 


80.8 


14.2 


80.5 


15.6 


80.2 


17.0 


79.9 


18.4 


79.6 


19.8 


83 


82.2 


11.6 


82.0 


13.0 


81.7 


14.4 


81.5 


15.8 


81.2 


17.3 


80.9 


18.7 


80.5 


20.1 


84 


83.2 


11.7 


83.0 


13.1 


82.7 


14.6 


82.5 


16.0 


82.2 


17.5 


81.8 


18.9 


81.5 


20.3 


85 


84.2 


11.8 


84.0 


13.3 


83.7 


14.8 


83.4 


16.2 


83.1 


17.7 


82.8 


19.1 


82.5 


20.6 


86 


85.2 


12.0 


84.9 


13.5 


84.7 


14.9 


84.4 


16.4 


84.1 


17.9 


83.8 


19.3 


83.4 


20.8 


87 


86.2 


12.1 


85.9 


13.6 


85.7 


15.1 


85.4 


16.6 


85.1 


18.1 


84.8 


19.6 


84.4 


21.0 


88 


87.1 


12.2 


86.9 


13.8 


86.7 


15.3 


86.4 


16.8 


86.1 


18.3 


85.7 


19.8 


85.4 


21.3 


89 


88.1 


12.4 


87.9 


13.9 


87.6 


15.5 


87.4 


17.0 


87.1 


18.5 


86.7 


20.0 


86.4 


21.5 


90 


89.1 


12.5 


88.9 


14.1 


88.6 


15.6 


88.3 


17.2 


88.0 


18.7 


87.7 


20.2 


87.3 


21.8 


91 


90.1 


12.7 


89.9 


14.2 


89.6 


15.8 


89.3 


17.4 


89.0 


18.9 


88.7 


20.5 


88.3 


22.0 


92 


91.1 


12.8 


90.9 


14.4 


90.6 


16.0 


90.3 


17.6 


90.0 


19.1 


89.6 


20.7 


89.3 


22.3 


93 


92.1 


12.9 


91.9 


14.5 


91.6 


16.1 


91.3 


17.7 


91.0 


19.3 


90.6 


20.9 


90.2 


22.5 


94 


93.1 


13.1 


92.8 


14.7 


92.6 


16.3 


92.3 


17.9 


91.9 


19.5 


91.6 


21.1 


91.2 


22.7 


95 


94.1 


13.2 


93.8 


14.9 


93.6 


16.5 


93.3 


18.1 


92.9 


19.8 


92.6 


21.4 


92.2 


23.0 


96 


95.1 


13.4 


94.8 


15.0 


94.5 


16.7 


94.2 


18.3 


93.9 


20.0 


93.5 


21.6 


93.1 


23.2 


97 


96.1 


13.5 


95.8 


15.2 


95.5 


16.8 


95.2 


18.5 


94.9 


20.2 


94.5 


21.8 


94.1 


23.5 


98 


97.0 


13.6 


96.8 


15.3 


96.5 


17.0 


96.2 


18.7 


95.9 


20.4 


95.5 


22.0 


95.1 


23.7 


99 


98.0 


13.8 


97.8 


15.5 


97.5 


17.2 


97.2 


18.9 


96.8 


20.6 


96.5 


22.3 


96.1 


24.0 


100 


99.0 


13.9 


98.8 


15.6 


98.5 


17.4 


98.2 


19.1 


97.8 


20.8 


97.4 


22.5 


97.0 


24.2 


600 


594.2 


83.5 


592. f 


93.8 


590.9 


104.2 


589.0 


114.5 


586.9 


124.7 


584.6 


135.0 


582.2 


145.1 


700 


693.3 


97.4 


691.3 


109.4 


689.5 


121.5 


687.1 


133.6 


684.7 


145.5 


682.1 


157.5 


679.2 


169.3 


800 


792.3 


111.4 


790.2 


125.1 


787.9 


139.0 


785.2 


152.6 


782.5 


166.3 


779.4 


180.0 


776.2 


193.6 


900 


891.3 


125.2 


888.8 


140.8 


886.3 


156.3 


883.3 


171.7 


880.2 


187.1 


S70.S 


202.4 


873.2 


217.7 




Dep 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 




(98, 262, 


(99, 261, 


(100, 260, 


(101, 259, 


(102, 258, 


(103, 257, 


(104, 256, 




278) 


279) 


280) 


281) 


282) 


283) 


284) 




1\ Pt. 82 


81 


80 


7 Pt. 79 


78 


77 


61 Pt. 76 



The 7-Pt. or 79 Courses are : E. by N., W. by N., E. by S., W. by S. 



158 



Table 1. Traverse Table 





15 


16 


IJPt. 17 


18 


19 


1J Pt, 20 




(165, 195, 


(164, 196, 


(163, 197, 


(162, 198, 


(161, 199, 


(160, 200, 


DlST. 


345) 


344) 


343) 


342) 


341) 


340) 




Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


1 


1.0 


0.3 


1.0 


0.3 


1.0 


0.3 


1.0 


0.3 


0.9 


0.3 


0.9 


0.3 


2 


1 Q 


0.5 


1.9 


0.6 


1.9 


0.6 


1.9 


0.6 


1.9 


0.7 


1.9 


07 


3 


2.9 


0.8 


2.9 


0.8 


2.9 


0.9 


2.9 


0.9 


2.8 


1.0 


2.8 


1.0 


4 


3.9 


1.0 


3.8 


1.1 


3.8 


1.2 


3.8 


1.2 


3.8 


1.3 


3.8 


1.4 


5 


4.8 


1.3 


4.8 


1.4 


4.8 


1.5 


4.8 


1.5 


4.7 


1.6 


4.7 


1.7 


6 


5.8 


1.6 


5.8 


1.7 


5.7 


1.8 


5.7 


1.9 


5.7 


2.0 


5.6 


2.1 


7 


6.8 


1.8 


6.7 


1.9 


6.7 


2.0 


6.7 


2.2 


6.6 


2.3 


6.6 


2.4 


8 


7.7 


2.1 


7.7 


2.2 


7.7 


2.3 


7.6 


2.5 


7.6 


2.6 


7.5 


2.7 


9 


8.7 


2.3 


8.7 


2.5 


8.6 


2.6 


8.6 


2.8 


8.5 


2.9 


8.5 


3.1 


10 


9.7 


2.6 


9.6 


2.8 


9.6 


2.9 


9.5 


3.1 


9.5 


3.3 


9.4 


3.4 


11 


10.6 


2.8 


10.6 


3.0 


10.5 


3.2 


10.5 


3.4 


10.4 


3.6 


10.3 


3.8 


12 


11.6 


3.1 


11.5 


3.3 


11.5 


3.5 


11.4 


3.7 


11.3 


3.9 


11.3 


4.1 


13 


12.6 


3.4 


12.5 


3.6 


12.4 


3.8 


12.4 


4.0 


12.3 


4.2 


12.2 


4.4 


14 


13.5 


3.6 


13.5 


3.9 


13.4 


4.1 


13.3 


4.3 


13.2 


4.6 


13.2 


4.8 


15 


14.5 


3.9 


14.4 


4.1 


14.3 


4.4 


14.3 


4.6 


14.2 


4.9 


14.1 


5.1 


16 


15.5 


4.1 


15.4 


4.4 


15.3 


4.7 


15.2 


4.9 


15.1 


5.2 


15.0 


5.5 


17 


16.4 


4.4 


16.3 


4.7 


16.3 


5.0 


16.2 


5.3 


16.1 


5.5 


16.0 


5.8 


18 


17.4 


4.7 


17.3 


5.0 


17.2 


5.3 


17.1 


5.6 


17.0 


5.9 


16.9 


6.2 


19 


18.4 


4.9 


18.3 


5.2 


18.2 


5.6 


18.1 


5.9 


18.0 


6.2 


17.9 


6.5 


20 


19.3 


5.2 


19.2 


5.5 


19.1 


5.8 


19.0 


6.2 


18.9 


6.5 


18.8 


6.8 


21 


20.3 


5.4 


20.2 


5.8 


20.1 


6.1 


20.0 


6.5 


19.9 


6.8 


19.7 


7.2 


22 


21.3 


5.7 


21.1 


6.1 


21.0 


6.4 


20.9 


6.8 


20.8 


7.2 


20.7 


7.5 


23 


22.2 


6.0 


22.1 


6.3 


22.0 


6.7 


21.9 


7.1 


21.7 


7.5 


21.6 


7.9 


24 


23.2 


6.2 


23.1 


6.6 


23.0 


7.0 


22.8 


7.4 


22.7 


7.8 


22.6 


8.2 


25 


24.1 


6.5 


24.0 


6.9 


23.9 


7.3 


23.8 


7.7 


23.6 


8.1 


23.5 


8.6 


26 


25 1 


6.7 


25.0 


7.2 


24.9 


7.6 


?47 


8.0 


24.6 


8.5 


24.4 


89 


27 


26.1 


7.0 


26.0 


7.4 


25.8 


7.9 


25.7 


8.3 


25.5 


8.8 


25.4 


9.2 


28 


27.0 


7.2 


26.9 


7.7 


26.8 


8.2 


26.6 


8.7 


26.5 


9.1 


26.3 


9.6 


29 


28.0 


7.5 


27.9 


8.0 


27.7 


8.5 


27.6 


9.0 


27.4 


9.4 


27.3 


9.9 


30 


29.0 


7.8 


28.8 


8.3 


28.7 


8.8 


28.5 


9.3 


28.4 


9.8 


28.2 


10.3 


31 


29.9 


8.0 


29.8 


8.5 


29.6 


9.1 


29.5 


9.6 


29.3 


10.1 


29.1 


10.6 


32 


30.9 


8.3 


30.8 


8.8 


30.6 


9.4 


30.4 


9.9 


30.3 


10.4 


30.1 


10.9 


33 


31.9 


8.5 


31.7 


9.1 


31.6 


9.6 


31.4 


10.2 


31.2 


10.7 


31.0 


11.3 


34 


32.8 


8.8 


32.7 


9.4 


32.5 


9.9 


32.3 


10.5 


32.1 


11.1 


31.9 


11.6 


35 


33.8 


9.1 


33.6 


9.6 


33.5 


10.2 


33.3 


10.8 


33.1 


11.4 


32.9 


12.0 


36 


34.8 


9.3 


34.6 


9.9 


34.4 


10.5 


34.2 


11.1 


34.0 


11.7 


33.8 


12.3 


37 


35.7 


9.6 


35.6 


10.2 


35.4 


10.8 


35.2 


11.4 


35.0 


12.0 


34.8 


12.7 


38 


36.7 


9.8 


36.5 


10.5 


36.3 


11.1 


36.1 


11.7 


35.9 


12.4 


35.7 


13.0 


39 


37.7 


10.1 


37.5 


10.7 


37.3 


11.4 


37.1 


12.1 


36.9 


12.7 


36.6 


13.3 


40 


38.6 


10.4 


38.5 


11.0 


38.3 


11.7 


38.0 


12.4 


37.8 


13.0 


37.6 


13.7 


41 


39.6 


10.6 


39.4 


11.3 


39.2 


12.0 


39.0 


12.7 


38.8 


13.3 


38.5 


14.0 


42 


40.6 


10.9 


40.4 


11.6 


40.2 


12.3 


39.9 


13.0 


39.7 


13.7 


39.5 


14.4 


43 


41.5 


11.1 


41.3 


11.9 


41.1 


12.6 


40.9 


13.3 


40.7 


14.0 


40.4 


14.7 


44 


42.5 


11.4 


42.3 


12.1 


42.1 


12.9 


41.8 


13.6 


41.6 


14.3 


41.3 


15.0 


45 


43.5 


11.6 


43.3 


12.4 


43.0 


13.2 


42.8 


13.9 


42.5 


14.7 


42.3 


15.4 


46 


44.4 


11.9 


44.2 


12.7 


44.0 


13.4 


43.7 


14.2 


43.5 


15.0 


43.2 


15.7 


47 


45.4 


12.2 


45.2 


13.0 


44.9 


13.7 


44.7 


14.5 


44.4 


15.3 


44.2 


16.1 


48 


46.4 


12.4 


46.1 


13.2 


45.9 


14.0 


45.7 


14.8 


45.4 


15.6 


45.1 


16.4 


49 


47.3 


12.7 


47.1 


13.5 


46.9 


14.3 


46.6 


15.1 


46.3 


16.0 


46.0 


16.8 


50 


48.3 


12.9 


48.1 


13.8 


47.8 


14.6 


47.6 


15.5 


47.3 


16.3 


47.0 


17.1 


100 


96.6 


25.9 


96.1 


27.6 


95.6 


29.2 


95.1 


30.9 


94.6 


32.6 


94.0 


34.2 


200 


193.2 


51.8 


192.3 


55.1 


191.3 


58.5 


190.2 


61.8 


189.1 


65.1 


187.9 


68.4 


300 


289.8 


77.6 


288.4 


82.7 


286.9 


87.7 


285.3 


92.7 


283.7 


97.7 


281.9 


102.6 


400 


386.3 


103.5 


384.5 


110.2 


382.5 


117.0 


380.4 


123.6 


378.2 


130.2 


375.9 


136.8 


500 


483.0 


129.4 


480.6 


137.8 


478.1 


146.2 


475.5 


154.5 


472.8 


162.8 


469.9 


171.0 




Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 




(105, 255, 


(106, 254, 


(107, 253, 


(108, 252, 


(109, 251, 


(110, 250, 




285) 


286) 


287) 


288) 


289) 


290) 




75 


74 


6 Pt, 73 


72 


71 


6i Pt. 70 



Table 1. Traverse Table 



159 





15 


16 


H Pt.l7 


18 


19 


If Pt. 20 




(165, 195, 


(164, 196, 


(163, 197, 


(162, 198, 


(161, 199, 


(160, 200, 


DlST. 


345) 


344) 


343) 


342) 


341) 


340) 




Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


51 


49.3 


13.2 


49.0 


14.1 


48.8 


14.9 


48.5 


15.8 


48.2 


16.6 


47.9 


17.4 


52 


50.2 


13.5 


50.0 


14.3 


49.7 


15.2 


49.5 


16.1 


49.2 


16.9 


48.9 


17.8 


53 


51.2 


13.7 


50.9 


14.6 


50.7 


15.5 


50.4 


16.4 


50.1 


17.3 


49.8 


18.1 


54 


52.2 


14.0 


51.9 


14.9 


51.6 


15.8 


51.4 


16.7 


51.1 


17.6 


50.7 


18.5 


55 


53.1 


14.2 


52.9 


15.2 


52.6 


16.1 


52.3 


17.0 


52.0 


17.9 


51.7 


18.8 


56 


54.1 


14.5 


53.8 


15.4 


53.6 


16.4 


53.3 


17.3 


52.9 


18.2 


52.6 


19.2 


57 


55.1 


14.8 


54.8 


15.7 


54.5 


16.7 


54.2 


17.6 


53.9 


18.6 


53.6 


19.5 


58 


56.0 


15.0 


55.8 


16.0 


55.5 


17.0 


55.2 


17.9 


54.8 


18.9 


54.5 


19.8 


50 


57.0 


15.3 


56.7 


16.3 


56.4 


17.2 


56.1 


18.2 


55.8 


19.2 


55.4 


20.2 


60 


58.0 


15.5 


57.7 


16.5 


57.4 


17.5 


57.1 


18.5 


56.7 


19.5 


56.4 


20.5 


61 


58.9 


15.8 


58.6 


16.8 


58.3 


17.8 


58.0 


18.9 


57.7 


19.9 


57.3 


20.9 


62 


59.9 


16.0 


59.6 


17.1 


59.3 


18.1 


59.0 


19.2 


58.6 


20.2 


58.3 


21.2 


63 


60.9 


16.3 


60.6 


17.4 


60.2 


18.4 


59.9 


19.5 


59.6 


20.5 


59.2 


21.5 


64 


61.8 


16.6 


61.5 


17.6 


61.2 


18.7 


60.9 


19.8 


60.5 


20.8 


60.1 


21.9 


65 


62.8 


16.8 


62.5 


17.9 


62.2 


19.0 


61.8 


20.1 


61.5 


21.2 


61.1 


22.2 


66 


63.8 


17.1 


63.4 


18.2 


63.1 


19.3 


62.8 


20.4 


62.4 


21.5 


62.0 


22.6 


67 


64.7 


17.3 


64.4 


18.5 


64.1 


19.6 


63.7 


20.7 


63.3 


21.8 


63.0 


22.9 


68 


65.7 


17.6 


65.4 


18.7 


65.0 


19.9 


64.7 


21.0 


64.3 


22.1 


63.9 


23.3 


69 


66.6 


17.9 


66.3 


19.0 


66.0 


20.2 


65.6 


21.3 


65.2 


22.5 


64.8 


23.6 


70 


67.6 


18.1 


67.3 


19.3 


66.9 


20.5 


66.6 


21.6 


66.2 


22.8 


65.8 


23.9 


71 


68.6 


18.4 


68.2 


19.6 


67.9 


20.8 


67.5 


21.9 


67.1 


23.1 


66.7 


24.3 


72 


69.5 


18.6 


69.2 


19.8 


68.9 


21.1 


68.5 


22.2 


68.1 


23.4 


67.7 


24.6 


73 


70.5 


18.9 


70.2 


20.1 


69.8 


21.3 


69.4 


22.6 


69.0 


23.8 


68.6 


25.0 


74 


71.5 


19.2 


71.1 


20.4 


70.8 


21.6 


70.4 


22.9 


70.0 


24.1 


69.5 


25.3 


75 


72.4 


19.4 


72.1 


20.7 


71.7 


21.9 


71.3 


23.2 


70.9 


24.4 


70.5 


25.7 


76 


73.4 


19.7 


73.1 


20.9 


72.7 


22.2 


72.3 


23.5 


71.9 


24.7 


71.4 


26.0 


77 


74.4 


19.9 


74.0 


21.2 


73.6 


22.5 


73.2 


23.8 


72.8 


25.1 


72.4 


26.3 


78 


75.3 


20.2 


75.0 


21.5 


74.6 


22.8 


74.2 


24.1 


73.8 


25.4 


73.3 


26.7 


79 


76.3 


20.4 


75.9 


21.8 


75.5 


23.1 


75.1 


24.4 


74.7 


25.7 


74.2 


27.0 


80 


77.3 


20.7 


76.9 


22.1 


76.5 


23.4 


76.1 


24.7 


75.6 


26.0 


75.2 


27.4 


81 


78.2 


21.0 


77.9 


22.3 


77.5 


23.7 


77.0 


25.0 


76.6 


26.4 


76.1 


27.7 


82 


79.2 


21.2 


78.8 


22.6 


78.4 


24.0 


78.0 


25.3 


77.5 


26.7 


77.1 


28.0 


83 


80.2 


21.5 


79.8 


22.9 


79.4 


24.3 


78.9 


25.6 


78.5 


27.0 


78.0 


28.4 


84 


81.1 


21.7 


80.7 


23.2 


80.3 


24.6 


79.9 


26.0 


79.4 


27.3 


78.9 


28.7 


85 


82.1 


22.0 


81.7 


23.4 


81.3 


24.9 


80.8 


26.3 


80.4 


27.7 


79.9 


29.1 


86 


83.1 


22.3 


82.7 


23.7 


82.2 


25.1 


81.8 


26.6 


81.3 


28.0 


80.8 


29.4 


87 


84.0 


22.5 


83.6 


24.0 


83.2 


25.4 


82.7 


26.9 


82.3 


28.3 


81.8 


29.8 


88 


85.0 


22.8 


84.6 


24.3 


84.2 


25.7 


83.7 


27.2 


83.2 


28.7 


82.7 


30.1 


80 


86.0 


23.0 


85.6 


24.5 


85.1 


26.0 


84.6 


27.5 


84.2 


29.0 


83.6 


30.4 


90 


86.9 


23.3 


86.5 


24.8 


86.1 


26.3 


85.6 


27.8 


85.1 


29.3 


84.6 


30.8 


01 


87.9 


23.6 


87.5 


25.1 


87.0 


26.6 


86.5 


28.1 


86.0 


29.6 


85.5 


31.1 


92 


88.9 


23.8 


88.4 


25.4 


88.0 


26.9 


87.5 


28.4 


87.0 


30.0 


86.5 


31.5 


93 


89.8 


24.1 


89.4 


25.6 


SS.9 


27.2 


88.4 


28.7 


87.9 


30.3 


87.4 


31.8 


94 


90.8 


24.3 


90.4 


25.9 


89.9 


27.5 


89.4 


29.0 


88.9 


30.6 


88.3 


32.1 


95 


91.8 


24.6 


91.3 


26.2 


90.8 


27.8 


90.4 


29.4 


89.8 


30.9 


89.3 


32.5 


96 


92.7 


24.8 


92.3 


26.5 


91.8 


28.1 


91.3 


29.7 


90.8 


31.3 


90.2 


32.8 


97 


93.7 


25.1 


93.2 


26.7 


92.8 


28.4 


92.3 


30.0 


91.7 


31.6 


91.2 


33.2 


98 


94.7 


25.4 


94.2 


27.0 


93.7 


28.7 


93.2 


30.3 


92.7 


31.9 


92.1 


33.5 


99 


95.6 


25.6 


95.2 


27.3 


94.7 


28.9 


94.2 


30.6 


93.6 


32.2 


93.0 


33.9 


100 


96.6 


25.9 


96.1 


27.6 


95.6 


29.2 


95.1 


30.9 


94.6 


32.6 


94.0 


34.2 


600 


579.5 


155.3 


576.8 


165.4 


573.8 


175.4 


570.6 


185.4 


567.3 


195.3 


563.8 


205.2 


700 


676.1 


181.1 


672.8 


193.0 


669.4 


204.6 


665.8 


216.3 


661.9 


227.9 


657.9 


239.4 


800 


772.7 


207.0 


769.0 


220.5 


765.0 


233.9 


760.8 


247.3 


756.5 


260.4 


751.8 


273.6 


900 


869.2 


232.9 


865.0 


248.0 


860.6 


263.1 


855.9 


278.1 


850.9 


_".)!'. '.1 


845.7 


307.8 




Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 




(105. 255, 


(106, 254, 


(107, 253, 


(108, 252, 


(109, 251, 


(110, 250, 




285) 


286) 


287) 


288) 


289) 


290) 




75 


74 


6i Pt. 73 


72 


71 


70 



160 



Table 1. Traverse Table 





21 


22 


2 Ft. 23 


24 


2| Ft. 25 


26 


DlST. 


(159, 201, 


(158, 202, 


(157, 203, 


(156, 204, 


(155, 205, 


(154, 206, 




339) 


338) 


337) 


336) 


886) 


334) 




Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


1 


0.9 


0.4 


0.9 


0.4 


0.9 


0.4 


0.9 


0.4 


0.9 


0.4 


0.9 


0.4 


2 


1.9 


0.7 


1.9 


0.7 


1.8 


0.8 


1.8 


0.8 


1.8 


0.8 


1.8 


0.9 


3 


2.8 


1.1 


2.8 


1.1 


2.8 


1.2 


2.7 


1.2 


2.7 


1.3 


2.7 


1.3 


4 


3.7 


1.4 


3.7 


1.5 


3.7 


1.6 


3.7 


1.6 


3.6 


1.7 


3.6 


1.8 


5 


4.7 


1.8 


4.6 


1.9 


4.6 


2.0 


4.6 


2.0 


4.5 


2.1 


4.5 


2.2 


6 


5.6 


2.2 


5.6 


2.2 


5.5 


2.3 


5.5 


2.4 


5.4 


2.5 


5.4 


2.6 


7 


6.5 


2.5 


6.5 


2.6 


6.4 


2.7 


6.4 


2.8 


6.3 


3.0 


6.3 


3.1 


8 


75 


2.9 


7.4 


3.0 


74 


3.1 


7.3 


3.3 


7.3 


3.4 


7.2 


35 


9 


8.4 


3.2 


8.3 


3.4 


8.3 


3.5 


8.2 


3.7 


8.2 


3.8 


8.1 


3.9 


10 


9.3 


3.6 


9.3 


3.7 


9.2 


3.9 


9.1 


4.1 


9.1 


4.2 


9.0 


4.4 


11 


10.3 


3.9 


10.2 


4.1 


10.1 


4.3 


10.0 


4.5 


10.0 


4.6 


9.9 


4.8 


12 


11.2 


4.3 


11.1 


4.5 


11.0 


4.7 


11.0 


4.9 


10.9 


5.1 


10.8 


5.3 


13 


12 1 


4.7 


12.1 


4.9 


12.0 


5.1 


11.9 


5.3 


11.8 


5.5 


11.7 


57 


14 


13.1 


5.0 


13.0 


5.2 


12.9 


5.5 


12.8 


5.7 


12.7 


5.9 


12.6 


6.1 


15 


14.0 


5.4 


13.9 


5.6 


13.8 


5.9 


13.7 


6.1 


13.6 


6.3 


13.5 


6.6 


16 


14.9 


5.7 


14.8 


6.0 


14.7 


6.3 


14.6 


6.5 


14.5 


6.8 


14.4 


7.0 


17 


15.9 


6.1 


15.8 


6.4 


15.6 


6.6 


15.5 


6.9 


15.4 


7.2 


15.3 


7.5 


18 


16.8 


6.5 


16.7 


6.7 


16.6 


7.0 


16.4 


7.3 


16.3 


7.6 


16.2 


7.9 


19 


17.7 


6.8 


17.6 


7.1 


17.5 


7.4 


17.4 


7.7 


17.2 


8.0 


17.1 


8.3 


20 


18.7 


7.2 


18.5 


7.5 


18.4 


7.8 


18.3 


8.1 


18.1 


8.5 


18.0 


8.8 


21 


19.6 


7.5 


19.5 


7.9 


19.3 


8.2 


19.2 


8.5 


19.0 


8.9 


18.9 


9.2 


22 


20.5 


7.9 


20.4 


8.2 


20.3 


8.6 


20.1 


8.9 


19.9 


9.3 


19.8 


9.6 


23 


21.5 


8.2 


21.3 


8.6 


21.2 


9.0 


21.0 


9.4 


20.8 


9.7 


20.7 


10.1 


24 


22.4 


8.6 


22.3 


9.0 


22.1 


9.4 


21.9 


9.8 


21.8 


10.1 


21.6 


10.5 


25 


23.3 


9.0 


23.2 


9.4 


23.0 


9.8 


22.8 


10.2 


22.7 


10.6 


22.5 


11.0 


26 


24.3 


9.3 


24.1 


9.7 


23.9 


10.2 


23.8 


10.6 


23.6 


11.0 


23.4 


11.4 


27 


25.2 


9.7 


25.0 


10.1 


24.9 


10.5 


24.7 


11.0 


24.5 


11.4 


24.3 


11.8 


28 


26.1 


10.0 


26.0 


10.5 


25.8 


10.9 


25.6 


11.4 


25.4 


11.8 


25.2 


12.3 


29 


27.1 


10.4 


26.9 


10.9 


26.7 


11.3 


26.5 


11.8 


26.3 


12.3 


26.1 


12.7 


30 


28.0 


10.8 


27.8 


11.2 


27.6 


11.7 


27.4 


12.2 


27.2 


12.7 


27.0 


13.2 


31 


28.9 


11.1 


28.7 


11.6 


28.5 


12.1 


28.3 


12.6 


28.1 


13.1 


27.9 


13.6 


32 


29.9 


11.5 


29.7 


12.0 


29.5 


12.5 


29.2 


13.0 


29.0 


13.5 


28.8 


14.0 


33 


30.8 


11.8 


30.6 


12.4 


30.4 


12.9 


30.1 


13.4 


29.9 


13.9 


29.7 


14.5 


34 


31.7 


12.2 


31.5 


12.7 


31.3 


13.3 


31.1 


13.8 


30.8 


14.4 


30.6 


14.9 


35 


32.7 


12.5 


32.5 


13.1 


32.2 


13.7 


32.0 


14.2 


31.7 


14.8 


31.5 


15.3 


36 


33.6 


12.9 


33.4 


13.5 


33.1 


14.1 


32.9 


14.6 


32.6 


15.2 


32.4 


15.8 


37 


34.5 


13.3 


34.3 


13.9 


34.1 


14.5 


33.8 


15.0 


33.5 


15.6 


33.3 


16.2 


38 


35 5 


13.6 


35.2 


14.2 


35.0 


14.8 


34.7 


15.5 


34.4 


16.1 


34.2 


167 


39 


36.4 


14.0 


36.2 


14.6 


35.9 


15.2 


35.6 


15.9 


35.3 


16.5 


35.1 


17.1 


40 


37.3 


14.3 


37.1 


15.0 


36.8 


15.6 


36.5 


16.3 


36.3 


16.9 


36.0 


17.5 


41 


38.3 


14.7 


38.0 


15.4 


37.7 


16.0 


37.5 


16.7 


37.2 


17.3 


36.9 


18.0 


42 


39.2 


15.1 


38.9 


15.7 


38.7 


16.4 


38.4 


17.1 


38.1 


17.7 


37.7 


18.4 


43 


40.1 


15.4 


39.9 


16.1 


39.6 


16.8 


39.3 


17.5 


39.0 


18.2 


38.6 


18.8 


44 


41.1 


15.8 


40.8 


16.5 


40.5 


17.2 


40.2 


17.9 


39.9 


18.6 


39.5 


19.3 


45 


42.0 


16.1 


41.7 


16.9 


41.4 


17.6 


41.1 


18.3 


40.8 


19.( 


40.4 


19.7 


46 


42.9 


16.5 


42.7 


17.2 


42.3 


18.0 


42.0 


18.7 


41.7 


19.4 


41.3 


20.2 


47 


43.9 


16.8 


43.6 


17.6 


43.3 


18.4 


42.9 


19.1 


42.6 


19.9 


42.2 


20.6 


48 


44.8 


17.2 


44.5 


18.0 


44.2 


18.8 


43.9 


19.5 


43.5 


20.3 


43.1 


21.0 


49 


45.7 


17.6 


45.4 


18.4 


45.1 


19.1 


44.8 


19.9 


44.4 


20.7 


44.0 


21.5 


50 


46.7 


17.9 


46.4 


18.7 


46.0 


19.5 


45.7 


20.3 


45.3 


21.1 


44.9 


21.9 


100 


93.4 


35.8 


92.7 


37.5 


92.1 


39.1 


91.4 


40.7 


90.6 


42.3 


89.9 


43.8 


200 


186.7 


71.7 


185.4 


74.9 


184.1 


78.1 


182.7 


81.3 


181.3 


84.5 


179.8 


87.7 


300 


280.1 


107.5 


278.2 


112.4 


276.2 


117.2 


274.1 


122.0 


271.9 


126.S 


269.6 


131.5 


400 


373.4 


143.4 


370.9 


149.8 


368.2 


156.3 


365.4 


162.7 


362.5 


169.0 


359.5 


175.4 


500 


466.8 


179.2 


463.6 


187.3 


460.2 


195.4 


456.8 


203.4 


453.1 


211.3 


449.4 


219.2 




Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 




(111, 249, 


(112, 248, 


(113. 247, 


(114, 246, 


(115, 245, 


(116, 244, 




291) 


292) 


293) 


294) 


295) 


296) 




69 


G Ft. 68 


67 


66 


5f Ft. 65 


64 



The 2-Pt. or 23 Courses are : N.N.E., N.N.W., S.S.E., S.S.W. 



Table 1. Traverse Table 



161 





21 


22 


2 Pt. 23 


24 


21Pt, 25 


26 




(159, 201, 


(158, 202, 


(157, 203, 


(156, 204, 


(155, 205, 


(154, 206, 


DOT. 


339) 


338) 


337) 


336) 


335) 


334) 




Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


51 


47.6 


18.3 


47.3 


19.1 


46.9 


19.9 


46.6 


20.7 


46.2 


21.6 


45.8 


22.4 


52 


48.5 


18.6 


48.2 


19.5 


47.9 


20.3 


47.5 


21.2 


47.1 


22.0 


46.7 


22.8 


53 


49.5 


19.0 


49.1 


19.9 


48.8 


20.7 


48.4 


21.6 


48.0 


22.4 


47.6 


23.2 


54 


50.4 


19.4 


50.1 


20.2 


49.7 


21.1 


49.3 


22.0 


48.9 


22.8 


48.5 


23.7 


55 


51.3 


19.7 


51.0 


20.6 


50.6 


21.5 


50.2 


22.4 


49.8 


23,2 


49.4 


24.1 


56 


52.3 


20.1 


51.9 


21.0 


51.5 


21.9 


51.2 


22.8 


50.8 


23.7 


50.3 


24.5 


57 


53.2 


20.4 


52.8 


21.4 


52.5 


22.3 


52.1 


23.2 


51.7 


24.J 


51.2 


25.0 


58 


54.1 


20.8 


53.8 


21.7 


53.4 


22.7 


53.0 


23.6 


52.6 


24.5 


52.1 


25.4 


59 


55.1 


21.1 


54.7 


22.1 


54.3 


23.1 


53.9 


24.0 


53.5 


24.9 


53.0 


25.9 


60 


56.0 


21.5 


55.6 


22.5 


55.2 


23.4 


54.8 


244 


54,4 


25.4 


53.9 


26.3 


61 


56.9 


21.9 


56.6 


22.9 


56.2 


23.8 


55.7 


24.8 


55.3 


25.8 


54.8 


26.7 


62 


57.9 


22.2 


57.5 


23.2 


57.1 


24.2 


56.6 


25.2 


56.2 


26.2 


55.7 


27.2 


63 


58.8 


22.6 


58.4 


23.6 


58.0 


24.6 


57.6 


25.6 


57.1 


26.6 


56.6 


27.6 


64 


59.7 


22.9 


59.3 


24.0 


58.9 


25.0 


58.5 


26.0 


58.0 


27.0 


57.5 


28.1 


65 


60.7 


23.3 


60.3 


24.3 


59.8 


25.4 


59.4 


26.4 


58.9 


27.5 


58.4 


28.5 


66 


61.6 


23.7 


61.2 


24.7 


60.8 


25.8 


60.3 


26.8 


59.8 


27.9 


59.3 


28.9 


67 


62.5 


24.0 


62.1 


25.1 


61.7 


26.2 


61.2 


27.3 


60.7 


28.3 


60.2 


29.4 


68 


63.5 


24.4 


63.0 


25.5 


62.6 


26.6 


62.1 


27.7 


61.6 


28.7 


61.1 


29.8 


69 


64.4 


24.7 


64.0 


25.8 


63.5 


27.0 


63.0 


28.1 


62.5 


29.2 


62.0 


30.2 


70 


65.4 


25.1 


64.9 


26.2 


64.4 


27.4 


63.9 


28.5 


63.4 


29.6 


62.9 


30.7 


71 


66.3 


25.4 


65.8 


26.6 


65.4 


27.7 


64.9 


28.9 


64.3 


30.0 


63.8 


31.1 


72 


67.2 


25.8 


66.8 


27.0 


66.3 


28.1 


65.8 


29.3 


65.3 


30.4 


64.7 


31.6 


73 


68.2 


26.2 


67.7 


27.3 


67.2 


28.5 


66.7 


29.7 


66.2 


30.9 


65.6 


32.0 


74 


69.1 


26.5 


68.6 


27.7 


68.1 


28.9 


67.6 


30.1 


67.1 


31.3 


66.5 


32.4 


75 


70.0 


26.9 


69.5 


28.1 


69.0 


29.3 


68.5 


30.5 


68.0 


31.7 


67.4 


32.9 


76 


71.0 


27.2 


70.5 


28.5 


70.0 


29.7 


69.4 


30.9 


68.9 


32.1 


68.3 


33.3 


77 


71.9 


27.6 


71.4 


28.8 


70.9 


30.1 


70.3 


31.3 


69.8 


32.5 


69.2 


33.8 


78 


72.8 


28.0 


72.3 


29.2 


71.8 


30.5 


71.3 


31.7 


70.7 


33'.0 


70fl 


34.2 


79 


73.0 


28.3 


73.2 


29.6 


72.7 


30.9 


72.2 


32.1 


71.6 


33.4 


71.0 


34.6 


80 


74.7 


28.7 


74.2 


30.0 


73.6 


31.3 


73.1 


32.5 


72.5 


33.8 


71.9 


35.1 


81 


75.6 


29.0 


75.1 


30.3 


74.6 


31.6 


74.0 


32.9 


73.4 


34.2 


72.8 


35.5 


82 


76.6 


29.4 


76.0 


30.7 


75.5 


32.0 


74.9 


33.4 


74.3 


34.7 


73.7 


35.9 


83 


77.5 


29.7 


77.0 


31.1 


76.4 


32.4 


75.8 


33.8 


75.2 


35.1 


74.6 


36.4 


84 


78.4 


30.1 


77.9 


31.5 


77.3 


32.8 


76.7 


34.2 


76.1 


35.5 


75.5 


36.8 


85 


79.4 


30.5 


78.8 


31.8 


78.2 


33.2 


77.7 


34.6 


77.0 


35.9 


76.4 


37.3 


86 


80.3 


30.8 


79.7 


32.2 


79.2 


33.6 


78.6 


35.0 


77.9 


36.3 


77.3 


37.7 


87 


81.2 


31.2 


80.7 


32.6 


80.1 


34.0 


79.5 


35.4 


78.8 


36.8 


78.2 


38.1 


88 


82.2 


31.5 


81.6 


33.0 


81.0 


34.4 


80.4 


35.8 


79.8 


37.2 


79.1 


38.6 


89 


83.1 


31.9 


82.5 


33.3 


81.9 


34.8 


81.3 


36.2 


80.7 


37.6 


80.0 


39.0 


90 


84.0 


32.3 


83.4 


33.7 


82.8 


35.2 


82.2 


36.6 


81.6 


38.0 


80.9 


39.5 


91 


85.0 


32.6 


84.4 


34.1 


83.8 


35.6 


83.1 


37.0 


82.5 


38.5 


81.8 


39.9 


92 


85.9 


33.0 


85.3 


34.5 


84.7 


35.9 


84.0 


37.4 


83.4 


38.9 


82.7 


40.3 


93 


86.8 


33.3 


86.2 


34.8 


85.6 


36.3 


85.0 


37.8 


84.3 


39.3 


83.6 


40.8 


94 


87.8 


33.7 


87.2 


35.2 


86.5 


36.7 


85.9 


38.2 


85.2 


39.7 


84.5 


41.2 


95 


88.7 


34.0 


88.1 


35.6 


87.4 


37.1 


86.8 


38.6 


86.1 


40.1 


85.4 


41.6 


96 


89.6 


34.4 


89.0 


36.0 


88.4 


37.5 


87.7 


39.0 


87.0 


40.6 


86.3 


42.1 


97 


90.6 


34.8 


89.9 


36.3 


89.3 


37.9 


88.6 


39.5 


87.9 


41.0 


87.2 


42.5 


98 


91.5 


35.1 


90.9 


36.7 


90.2 


38.3 


89.5 


39.9 


88.8 


41.4 


88.1 


43.0 


99 


92.4 


35.5 


91.8 


37.1 


91.1 


38.7 


90.4 


40.3 


89.7 


41.8 


89.0 


43.4 


100 


93.4 


35.8 


92.7 


37.5 


92.1 


39.1 


91.4 


40.7 


90.6 


42.3 


89.9 


43.8 


600 


560.1 


215.0 


556.3 


224.8 


552.3 


234.4 


548.1 


244.0 


543.8 


253.6 


539.3 


263.0 


700 


653.6 


250.8 


649.1 


262.2 


644.3 


273.5 


639.5 


284.7 


634.5 


295.8 


629.2 


306.8 


800 


746.9 


286.7 


741.8 


299.7 


736.4 


312.6 


730.8 


325.4 


725.1 


338.1 


719.1 


350.6 


900 


840.3 


322.5 


s:i 1 ..', 


337.1 


828.3 


351.7 


822.1 


:;<;r,. i) 


815.6 


:;,so.:; 


808.9 


394.5 




Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lafc. 


Dep. 


Lat. 




(111, 249, 


(112, 248, 


(113, 247, 


(114, 246, 


(115, 245, 


(116, 244, 




291) 


292) 


293) 


294) 


295) 


296) 




69 


6 Pt. 68 


67 


66 


5| Pt.65 


64 



The 6-Pt. or 68 Courses are : E.N.E., W.N.W., E.S.E., W.S.W. 



162 



Table 1. Traverse Table 





27 


2| Pt. 28 


29 


30 


2 f Pt. 31 


32 




(153, 207, 


(152, 208, 


(151, 209, 


(150, 210, 


(149, 211, 


(148, 212, 


DIST. 


333) 


332) 


331) 


330) 


329) 


328) 




Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


1 


0.9 


0.5 


0.9 


0.5 


0.9 


0.5 


0.9 


0.5 


0.9 


0.5 


0.8 


0.5 


2 


1.8 


0.9 


1.8 


0.9 


1.7 


1.0 


1.7 


1.0 


1.7 


1.0 


1.7 


1.1 


3 


2.7 


1.4 


2.6 


1.4 


2.6 


1.5 


2.6 


1.5 


2.6 


1.5 


2.5 


1.6 


4 


3.6 


1.8 


3.5 


1.9 


3.5 


1.9 


3.5 


2.0 


3.4 


2.1 


3.4 


2.1 


5 


4.5 


2.3 


4.4 


2.3 


4.4 


2.4 


4.3 


2.5 


4.3 


2.6 


4.2 


2.6 


6 


5.3 


2.7 


5.3 


2.8 


5.2 


2.9 


5.2 


3.0 


5.1 


3.1 


5.1 


3.2 


7 


6.2 


3.2 


6.2 


3.3 


6.1 


3.4 


6.1 


3.5 


6.0 


3.6 


5.9 


3.7 


8 


7.1 


3.6 


7.1 


3.8 


7.0 


3.9 


6.9 


4.0 


6.9 


4.1 


6.8 


4.2 


9 


8.0 


4.1 


7.9 


4.2 


7.9 


4.4 


7.8 


4.5 


7.7 


4.6 


7.6 


4.8 


10 


8.9 


4.5 


8.8 


4.7 


8.7 


4.8 


8.7 


5.0 


8.6 


5.2 


8.5 


5.3 


11 


9.8 


5.0 


9.7 


5.2 


9.6 


5.3 


9.5 


5.5 


9.4 


5.7 


9.3 


5.8 


12 


10.7 


5.4 


10.6 


5.6 


10.5 


5.8 


10.4 


6.0 


10.3 


6.2 


10.2 


6.4 


13 


11.6 


5.9 


11.5 


6.1 


11.4 


6.3 


11.3 


6.5 


11.1 


6.7 


11.0 


6.9 


14 


12.5 


6.4 


12.4 


6.6 


12.2 


6.8 


12.1 


7.0 


12.0 


7.2 


11.9 


7.4 


15 


13.4 


6.8 


13.2 


7.0 


13.1 


7.3 


13.0 


7.5 


12.9 


7.7 


12.7 


7.9 


16 


14.3 


. 7.3 


14.1 


7.5 


14.0 


7.8 


13.9 


8.0 


13.7 


8.2 


13.6 


8.5 


17 


15.1 


7.7 


15.0 


8.0 


14.9 


8.2 


14.7 


8.5 


14.6 


8.8 


14.4 


9.0 


18 


16.0 


8.2 


15.9 


8.5 


15.7 


8.7 


15.6 


9.0 


15.4 


9.3 


15.3 


9.5 


19 


16.9 


8.6 


16.8 


8.9 


16.6 


9.2 


16.5 


9.5 


16.3 


9.8 


16.1 


10.1 


20 


17.8 


9.1 


17.7 


9.4 


17.5 


9.7 


17.3 


10.0 


17.1 


10.3 


17.0 


10.6 


21 


18.7 


9.5 


18.5 


9.9 


18.4 


10.2 


18.2 


10.5 


18.0 


10.8 


17.8 


11.1 


22 


19.6 


10.0 


19.4 


10.3 


19.2 


10.7 


19.1 


11.0 


18.9 


11.3 


18.7 


11.7 


23 


20.5 


10.4 


20.3 


10.8 


20.1 


11.2 


19.9 


11.5 


19.7 


11.8 


19.5 


12.2 


24 


21.4 


10:9 


21:2 


11.3 


21.0 


11.6 


20.8 


12.0 


20.6 


12.4 


20.4 


12.7 


25 


22.3 


11.3 


22.1 


11.7 


21.9 


12.1 


21.7 


12.5 


21 A 


12.9 


21.2 


13.2 


26 


23.2 


11.8 


23.0 


12.2 


22.7 


12.6 


22.5 


13.0 


22.3 


13.4 


22.0 


13.8 


27 


24.1 


12.3 


23.8 


12.7 


23.6 


13.1 


23.4 


13.5 


23.1 


13.9 


22.9 


14.3 


28 


24.9 


12.7 


24.7 


13.1 


24.5 


13.6 


24.2 


14.0 


24.0 


14.4 


23.7 


14.8 


29 


25.8 


13.2 


25.6 


13.6 


25.4 


^A 1 


25.1 


14.5 


24.9 


14.9 


24.6 


15.4 


30 


26.7 


13.6 


26.5 


14.1 






26.0 


15.0 


25.7 


15.5 


25.4 


15.9 


31 


27.6 


14.1 


27.4 


14.6 


27.1 


15.0 


26.8 


15.5 


26.6 


16.0 


26.3 


16.4 


32 


28.5 


14.5 


28.3 


15.0 


28.0 


15.5 


27.7 


16.0 


27.4 


16.5 


27.1 


17.0 


33 


29.4 


15.0 


29.1 


15.5 


28.9 


16.0 


28.6 


16.5 


28.3 


17.0 


28.0 


17.5 


34 


30.3 


15.4 


30.0 


16.0 


29.7 


16.5 


29.4 


17.0 


29.1 


17.5 


28.8 


18.0 


35 


31.2 


15.9 


30.9 


16.4 


30.6 


17.0 


30.3 


17.5 


30.0 


18.0 


29.7 


18.5 


36 


32.1 


16.3 


31.8 


16.9 


31.5 


17.5 


31.2 


18.0 


30.9 


18.5 


30.5 


19.1 


37 


33.0 


16.8 


32.7 


17.4 


32.4 


17.9 


32.0 


18.5 


31.7 


19.1 


31.4 


19.6 


38 


33.9 


17.3 


33.6 


17.8 


33.2 


18.4 


32.9 


19.0 


32.6 


19.6 


32.2 


20.1 


39 


34.7 


17.7 


34.4 


18.3 


34.1 


18.9 


33.8 


19.5 


33.4 


20.1 


33.1 


20.7 


40 


35.6 


18.2 


35.3 


18.8 


35.0 


19.4 


34.6 


20.0 


34.3 


20.6 


33.9 


21.2 


41 


36.5 


18.6 


36.2 


19.2 


35.9 


19.9 


35.5 


20.5 


35.1 


21.1 


34.8 


21.7 


42 


37.4 


19.1 


37.1 


19.7 


36.7 


20.4 


36.4 


21.0 


36.0 


21.6 


35.6 


22.3 


43 


38.3 


19.5 


38.0 


20.2 


37.6 


20.8 


37.2 


21.5 


36.9 


22.1 


36.5 


22.8 


44 


39.2 


20.0 


38.8 


20.7 


38.5 


21.3 


38.1 


22.0 


37.7 


22.7 


37.3 


23.3 


45 


40.1 


20.4 


39.7 


21.1 


39.4 


21.8 


39.0 


22.5 


38.6 


23.? 


38.2 


23.8 


46 


41.0 


20.9 


40.6 


21.6 


40.2 


22.3 


39.8 


23.0 


39.4 


23.7 


39.0 


24.4 


47 


41.9 


21.3 


41.5 


22.1 


41.1 


22.8 


40.7 


23.5 


40.3 


24.2 


39.9 


24.9 


48 


42.8 


21.8 


42.4 


22.5 


42.0 


23.3 


41.6 


24.0 


41.1 


24.7 


40.7 


25.4 


49 


43.7 


22.2 


43.3 


23.0 


42.9 


23.8 


42.4 


24.5 


42.0 


25.2 


41.6 


26.0 


50 


44.6 


22.7 


44.1 


23.5 


43.7 


24.2 


43.3 


25.0 


42.9 


25.8 


42.4 


26.5 


100 


89.1 


45.4 


88.3 


46.9 


87.5 


48.5 


86.6 


50.0 


85.7 


51.5 


84.8 


53.0 


200 


178.2 


90.8 


176.6 


93.9 


174.9 


97.0 


173.2 


100.0 


171.4 


103.0 


169.6 


106.0 


300 


267.3 


136.2 


264.9 


140.8 


262.4 


145.4 


259.8 


150.0 


257.1 


154.5 


254.4 


159.0 


400 


356.4 


181.6 


353.1 


187.8 


349.8 


193.9 


346.4 


200.0 


342.9 


206.0 


339.2 


211.9 


500 


445.5 


227.0 


441.5 


234.7 


t:57.:^ 


242.4 


433.0 


250.0 


428.6 


257.5 


424.0 


265.0 




Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 




(117, 243, 


(118, 242, 


(119, 241, 


(120, 240, 


(121, 239, 


(122, 238, 




297) 


298) 


299) 


300) 


301) 


302) 




63 


5Pt. 62 


61 


60 


5| Pt. 59 


58 



Table 1. Traverse Table 



163 





27 


2i Pt. 28 


29 


30 


2J Pt. 31 


32 




(153, 207, 


(152, 208, 


(151, 209, 


(150, 210, 


(149, 21 1, 


(148, 212, 


DlST. 


333) 


332) 


331) 


330) 


329) 


328) 




Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


51 


45.4 


23.2 


45.0 


23.9 


44.6 


24.7 


44.2 


25.5 


43.7 


26.3 


43.3 


27.0 


52 


46.3 


23.6 


45.9 


24.4 


45.5 


25.2 


45.0 


26.0 


44.6 


26.8 


44.1 


27.6 


53 


47.2 


24.1 


46.8 


24.9 


46.4 


25.7 


45.9 


26.5 


45.4 


27.3 


44.9 


28.1 


54 


48.1 


24'.5 


47.7 


25.4 


47.2 


26.2 


46.8 


27.0 


46.3 


27.8 


45.8 


28.6 


55 


49.0 


25.0 


48.6 


25.8 


48.1 


26.7 


47.6 


27.5 


47J 


28.3 


46.6 


29.1 


56 


49.9 


25.4 


49.4 


26.3 


49.0 


27.1 


48.5 


28.0 


48.0 


28.8 


47.5 


29.7 


57 


50.8 


25.9 


50.3 


26.8 


49.9 


27.6 


49.4 


28.5 


48.9 


29.4 


48.3 


30.2 


58 


51.7 


26.3 


51.2 


27.2 


50.7 


28.1 


50.2 


29.0 


49.7 


29.9 


49.2 


30.7 


59 


52.6 


26.8 


52.1 


27.7 


51.6 


28.6 


51.1 


29.5 


50.6 


30.4 


50.0 


31.3 


60 


53.5 


27.2 


53.0 


28.2 


52.5 


29.1 


52.0 


30.0 


51.4 


30.9 


50.9 


31.8 


61 


54.4 


27.7 


53.9 


28.6 


53.4 


29.6 


52.8 


30.5 


52.3 


31.4 


51.7 


32.3 


62 


55.2 


28.1 


54.7 


29.1 


54.2 


30.1 


53.7 


31.0 


53.1 


31.9 


52.6 


32.9 


63 


56.1 


28.6 


55.6 


29.6 


55.1 


30.5 


54.6 


31.5 


54.0 


32.4 


53.4 


33.4 


64 


57.0 


29.1 


56.5 


30.0 


56.0 


31.0 


55.4 


32.0 


54.9 


33.0 


54.3 


33.9 


65 


57.9 


29.5 


57.4 


30.5 


56.9 


31.5 


56.3 


32.5 


55.7 


33.5 


55.1 


34.4 


66 


58.8 


30.0 


58.3 


31.0 


57.7 


32.0 


57.2 


33.0 


56.6 


34.0 


56.0 


35.0 


67 


59.7 


30.4 


59.2 


31.5 


58.6 


32.5 


58.0 


33.5 


57.4 


34.5 


56.8 


35.5 


68 


60.6 


30.9 


60.0 


31.9 


59.5 


33.0 


58.9 


34.0 


58.3 


35.0 


57.7 


36.0 


69 


61.5 


31.3 


60.9 


32.4 


60.3 


33.5 


59.8 


.34.5 


59.1 


35.5 


58.5 


36.6 


70 


62.4 


31.8 


61.8 


32.9 


61.2 


33.9 


60.6 


35.0 


60.0 


36.1 


59.4 


37.1 


71 


63.3 


32.2 


62.7 


33.3 


62.1 


34.4 


61.5 


35.5 


60.9 


36.6 


60.2 


37.6 


72 


64.2 


32.7 


63.6 


33.8 


63.0 


34.9 


62.4 


36.0 


61.7 


37.1 


61.1 


38.2 


73 


65.0 


33.1 


64.5 


34.3 


63.8 


35.4 


63.2 


36.5 


62.6 


37.6 


61.9 


38.7 


74 


65.9 


33.6 


65.3 


34.7 


64.7 


35.9 


64.1 


37.0 


63.4 


38.1 


62.8 


39.2 


75 


66.8 


34.0 


66.2 


35.2 


65.6 


36.4 


65.0 


37.5 


64.3 


38.6 


63.6 


39.7 


76 


67.7 


34.5 


67.1 


35.7 


66.5 


36.8 


65.8 


38.0 


65.1 


39.1 


64.5 


40.3 


77 


68.6 


35.0 


68.0 


36.1 


67.3 


37.3 


66.7 


38.5 


66.0 


39.7 


65.3 


40.8 


78 


69.5 


35.4 


68.9 


36.6 


68.2 


37.8 


67.5 


39.0 


66.9 


40.2 


66.1 


41.3 


79 


70.4 


35.9 


69.8 


37.1 


69.1 


38.3 


68.4 


39.5 


67.7 


40.7 


67.0 


41.9 


80 


71.3 


36.3 


70.6 


37.6 


70.0 


38.8 


69.3 


40.0 


68.6 


41.2 


67.8 


42.4 


81 


72.2 


36.8 


71.5 


38.0 


70.8 


39.3 


70.1 


40.5 


69.4 


41.7 


68.7 


42.9 


82 


73.1 


37.2 


72.4 


38.5 


71.7 


39.8 


71.0 


41.0 


70.3 


42.2 


69.5 


43.5 


83 


74.0 


37.7 


73.3 


39.0 


72.6 


40.2 


71.9 


41.5 


71.1 


42.7 


70.4 


44.0 


84 


74.8 


38.1 


74.2 


39.4 


73.5 


40.7 


Z2.7 


42.0 


72.0 


43.3 


71.2 


44.5 


85 


75.7 


38.6 


75.1 


39.9 


74.3 


41.2 


73.6 


42.5 


72.9 


43.8 


72.1 


45.0 


86 


76.6 


39.0 


75.9 


40.4 


75.2 


41.7 


74.5 


43.0 


73.7 


44.3 


72.9 


45.6 


87 


77.5 


39.5 


76.8 


40.8 


76.1 


42.2 


75.3 


43.5 


74.6 


44.8 


73.8 


46.1 


88 


78.4 


40.0 


77.7 


41.3 


77.0 


42.7 


76.2 


44.0 


75.4 


45.3 


74.6 


46.6 


89 


79.3 


40.4 


78.6 


41.8 


77.8 


43.1 


77.1 


44.5 


76.3 


45.8 


75.5 


47.2 


90 


80.2 


40.9 


79.5 


42.3 


78.7 


43.6 


77.9 


45.0 


77.1 


46.4 


76.3 


47.7 


91 


81.1 


41.3 


80.3 


42.7 


79.6 


44.1 


78.8 


45.5 


78.0 


46.9 


77.2 


48.2 


92 


82.0 


41.8 


81.2 


43.2 


80.5 


44.6 


79.7 


46.0 


78.9 


47.4 


78.0 


48.8 


93 


82.9 


42.2 


82.1 


43.7 


81.3 


45.1 


80.5 


46.5 


79.7 


47.9 


78.9 


49.3 


94 


83.8 


42.7 


83.0 


44.1 


82.2 


45.6 


81.4 


47.0 


80.6 


48.4 


79.7 


49.8 


95 


84.6 


43.1 


83.9 


44.6 


83.1 


46.1 


82.3 


47.5 


81.4 


48.9 


80.6 


50.3 


96 


85.5 


43.6 


84.8 


45.1 


84.0 


46.5 


83.1 


48.0 


82.3 


49.4 


81.4 


50.9 


97 


86.4 


44.0 


85.6 


45.5 


84.8 


47.0 


84.0 


48.5 


83.1 


50.0 


82.3 


51.4 


98 


87.3 


44.5 


86.5 


46.0 


85.7 


47.5 


84.9 


49.0 


84.0 


50.5 


83.1 


51.9 


99 


88.2 


44.9 


87.4 


46.5 


86.6 


48.0 


85.7 


49.5 


84.9 


51.0 


84.0 


52.5 


100 


89.1 


45.4 


88.3 


46.9 


87.5 


48.5 


86.6 


50.0 


85.7 


51.5 


84.8 


53.0 


600 


5346 


272.4 


529.8 


281.7 


524.8 


290.9 


519.6 


300.0 


514.3 


309.0 


508.8 


3180 


700 


623.7 


317.8 


618.0 


328.6 


612.2 


339.4 


606.1 


350.0 


600.1 


360.4 


593.6 


371.0 


800 


712.9 


363.2 


706.3 


375.6 


699.7 


387.9 


692.8 


400.0 


(isn.s 


412.0 


678.4 


423.9 


900 


801.9 


408.5 


794.5 


422.5 


787.0 


436.3 


779.3 


450.0 


771.4 


403.4 


763.2 


476.8 




Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 




(117, 243, 


(118, 242, 


(119, 241. 


(120, 240 


(121, 239, 


(122, 238, 




297) 


298) 


299) 


300) 


301) 


302) 


" 


63 


5i Pt. 62 


61 


60 


5i Pt. 59 


58 



164 



Table 1. Traverse Table 





33 


3 Pt. 34 


35 


36 


3J Pt. 37 


38 




(147, 213, 


(146, 214, 


(145, 215, 


(144, 216, 


(143, 217, 


(142, 218, 


DlST. 


327) 


326) 


325) 


324) 


323) 


322) 




Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


1 


0.8 


0.5 


0.8 


0.6 


0.8 


0.6 


0.8 


0.6 


0.8 


0.6 


0.8 


0.6 


2 


1.7 


1.1 


1.7 


1.1 


1.6 


1.1 


1.6 


1.2 


1.6 


1.2 


1.6 


1.2 


3 


2.5 


1.6 


2.5 


1.7 


2.5 


1.7 


2.4 


1.8 


2.4 


1.8 


2.4 


1.8 


4 


3.4 


2.2 


3.3 


2.2 


3.3 


2.3 


3.2 


2.4 


3.2 


2.4 


3.2 


2.5 


5 


4.2 


2.7 


4.1 


2.8 


4.1 


2.9 


4.0 


2.9 


4.0 


3.0 


3.9 


3.1 


6 


5.0 


3.3 


5.0 


3.4 


4.9 


3.4 


4.9 


3.5 


4.8 


3.6 


4.7 


3.7 


7 


5.9 


3.8 


5.8 


3.9 


5.7 


4.0 


5.7 


4.1 


5.6 


4.2 


5.5 


4.3 


8 


6.7 


4.4 


6.6 


4.5 


6.6 


4.6 


6.5 


4.7 


6.4 


4.8 


6.3 


4.9 


9 


7.5 


4.9 


7.5 


5.0 


7.4 


5.2 


7.3 


5.3 


7.2 


5.4 


7.1 


5.5 


10 


8.4 


5.4 


8.3 


5.6 


8.2 


5.7 


8.1 


5.9 


8.0 


6.0 


7.9 


6.2 


11 


9.2 


6.0 


9.1 


6.2 


9.0 


6.3 


8.9 


6.5 


8.8 


6.6 


8.7 


6.8 


12 


10.1 


6.5 


9.9 


6.7 


9.8 


6.9 


9.7 


7.1 


9.6 


7.2 


9.5 


7.4 


13 


10 Q 


7.1 


10.8 


7.3 


10.6 


7.5 


10.5 


7.6 


10.4 


7.8 


10.2 


80 


14 


11.7 


7.6 


11.6 


7.8 


11.5 


8.0 


11.3 


8.2 


11.2 


8.4 


11.0 


8.6 


15 


12.6 


8.2 


12.4 


8.4 


12.3 


8.6 


12.1 


8.8 


12.0 


9.0 


11.8 


9.2 


16 


13.4 


8.7 


13.3 


8.9 


13.1 


9.2 


12.9 


9.4 


12.8 


9.6 


12.6 


9.9 


17 


14.3 


9.3 


14.1 


9.5 


13.9 


9.8 


13.8 


10.0 


13.6 


10.2 


13.4 


10.5 


18 


15.1 


9.8 


14.9 


10.1 


14.7 


10.3 


14.6 


10.6 


14.4 


10.8 


14.2 


11.1 


19 


15.9 


10.3 


15.8 


10.6 


15.6 


10.9 


15.4 


11.2 


15.2 


11.4 


15.0 


11.7 


20 


16.8 


10.9 


16.6 


11.2 


16.4 


11.5 


16.2 


11.8 


16.0 


12.0 


15.8 


12.3 


21 


17.6 


11.4 


17.4 


11.7 


17.2 


12.0 


17.0 


12.3 


16.8 


12.6 


16.5 


12.9 


22 


18.5 


12.0 


18.2 


12.3 


18.0 


12.6 


17.8 


12.9 


17.6 


13.2 


17.3 


13.5 


23 


19.3 


12.5 


19.1 


12.9 


18.8 


13.2 


18.6 


13.5 


18.4 


13.8 


18.1 


14.2 


24 


20.1 


13.1 


19.9 


13.4 


19.7 


13.8 


19.4 


14.1 


19.2 


14.4 


18.9 


14.8 


25 


21.0 


13.6 


20.7 


14.0 


20.5 


14.3 


20.2 


14.7 


20.0 


15.0 


19.7 


15.4 


26 


21.8 


14.2 


21.6 


14.5 


21.3 


14.9 


21.0 


15.3 


20.8 


15.6 


20.5 


16.0 


27 


22.6 


14.7 


22.4 


15.1 


22.1 


15.5 


21.8 


15.9 


21.6 


16.2 


21.3 


16.6 


28 


23.5 


15.2 


23.2 


15.7 


22.9 


16.1 


22.7 


16.5 


22.4 


16.9 


22.1 


17.2 


29 


24.3 


15.8 


24.0 


16.2 


23.8 


16.6 


23.5 


17.0 


23.2 


17.5 


22.9 


17.9 


30 


25.2 


16.3 


24.9 


16.8 


24.6 


17.2 


24.3 


17.6 


24.0 


18.1 


23.6 


18.5 


31 


26.0 


16.9 


25.7 


17.3 


25.4 


17.8 


25.1 


18.2 


24.8 


18.7 


24.4 


19.1 


32 


26.8 


17.4 


26.5 


17.9 


26.2 


18.4 


25.9 


18.8 


25.6 


19.3 


25.2 


19.7 


33 


27.7 


18.0 


27.4 


18.5 


27.0 


18.9 


26.7 


19.4 


26.4 


19.9 


26.0 


20.3 


34 


28.5 


18.5 


28.2 


19.0 


27.9 


19.5 


27.5 


20.0 


27.2 


20.5 


26.8 


20.9 


35 


29.4 


19.1 


29.0 


19.6 


28.7 


20.1 


28.3 


20.6 


28.0 


21.1 


27.6 


21.5 


36 


30? 


19.6 


29.8 


20.1 


29.5 


20.6 


29.1 


21.2 


28.8 


21.7 


28.4 


??,2 


37 


31.0 


20.2 


30.7 


20.7 


30.3 


21.2 


29.9 


21.7 


29.5 


22.3 


29.2 


22.8 


38 


31.9 


20.7 


31.5 


21.2 


31.1 


21.8 


30.7 


22.3 


30.3 


22.9 


29.9 


23.4 


39 


32.7 


21.2 


32.3 


21.8 


31.9 


22.4 


31.6 


22.9 


31.1 


23.5 


30.7 


24.0 


40 


33.5 


21.8 


33.2 


22.4 


32.8 


22.9 


32.4 


23.5 


31.9 


24.1 


31.5 


24.6 


41 


34.4 


22.3 


34.0 


22.9 


33.6 


23.5 


33.2 


24.1 


32.7 


24.7 


32.3 


25.2 


42 


35.2 


22.9 


34.8 


23.5 


34.4 


24.1 


34.0 


24.7 


33.5 


25.3 


33.1 


25.9 


43 


36.1 


23.4 


35.6 


24.0 


35.2 


24.7 


34.8 


25.3 


34.3 


25.9 


33.9 


26.5 


44 


36.9 


24.0 


36.5 


24.6 


36.0 


25.2 


35.6 


25.9 


35.1 


26.5 


34.7 


27.1 


45 


37.7 


24.5 


37.3 


25.2 


36.9 


25.8 


36.4 


26.5 


35.9 


27.1 


35.5 


27.7 


46 


38.6 


25.1 


38.1 


25.7 


37.7 


26.4 


37.2 


27.0 


36.7 


27.7 


36.2 


28.3 


47 


39.4 


25.6 


39.0 


26.3 


38.5 


27.0 


38.0 


27.6 


37.5 


28.3 


37.0 


28.9 


48 


40.3 


26.1 


39.8 


26.8 


39.3 


27.5 


38.8 


28.2 


38.3 


28.9 


37.8 


29.6 


49 


41.1 


26.7 


40.6 


27.4 


40.1 


28.1 


39.6 


28.8 


39.1 


29.5 


38.6 


30.2 


50 


41.9 


27.2 


41.5 


28.0 


41.0 


28.7 


40.5 


29.4 


39.9 


30.1 


39.4 


30.8 


100 


83.9 


54.5 


82.9 


55.9 


81.9 


57.4 


80.9 


58.8 


79.9 


60.2 


78.8 


61.6 


200 


167.7 


108.9 


165.8 


111.8 


163.8 


114.7 


161.8 


117.6 


159.7 


120.4 


157.6 


123.1 


300 


251.6 


163.4 


248.7 


167.8 


245.7 


172.1 


242.7 


176.3 


239.6 


180.5 


236.4 


184.7 


400 


335.5 


217.8 


331.6 


223.7 


327.7 


229.4 


323.6 


235.1 


319.4 


240.7 


315.2 


246.3 


500 


419.3 


272.3 


414.5 


279.6 


409.6 


286.8 


404.5 


293.9 


399.3 


300.9 


394.0 


307.8 




Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 




(123, 237, 


(124, 236, 


(125, 235, 


(126, 234, 


(127, 233, 


(128, 232, 




303) 


304) 


305) 


306) 


307) 


308) 




57 


5 Pt. 56 


55 


54 


4fPt. 53 


52 



The 3-Pt. or 34 Courses are : N.E. by N., N.W. by N., S.E. by S., S.W. by S, 



Table 1. Traverse Table 



165 





33 


3 Pt. 34 


35 


36 


3iPt. 37 


38 




(147, 213, 


5(146, 214, 


(145, 215, 


(144, 216, 


(143, 217, 


(142, 218, 


DlST. 


827) 


326) 


325) 


324) 


323) 


322) 




Lat. 


Dep. 


iLat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


51 


42.8 


27.8 


42.3 


28.5 


41.8 


29.3 


41.3 


30.0 


40.7 


30.7 


40.2 


31.4 


52 


43.6 


28.3 


43.1 


29.1 


42.6 


29.8 


42.1 


30.6 


41.5 


31.3 


41.0 


32.0 


53 


44.4 


28.9 


43.9 


29.6 


43.4 


30.4 


42.9 


31.2 


42.3 


31.9 


41.8 


32.6 


54 


45.3 


29.4 


44.8 


30.2 


44.2 


31.0 


43.7 


31.7 


43.1 


32.5 


42.6 


33.2 


55 


46.1 


30.0 


45.6 


30.8 


45.1 


31.5 


44.5 


32.3 


43.9 


33.1 


43.3 


33.9 


56 


47.0 


30.5 


46.4 


31.3 


45.9 


32.1 


45.3 


32.9 


44.7 


33.7 


44.1 


34.5 


57 


47.8 


31.0 


47.3 


31.9 


46.7 


32.7 


46.1 


33.5 


45.5 


34.3 


44.9 


35.1 


58 


48.6 


31.6 


48.1 


32.4 


47.5 


33.3 


46.9 


34.1 


46.3 


34.9 


45.7 


35.7 


59 


49.5 


32.1 


48.9 


33.0 


48.3 


33.8 


47.7 


34.7 


47.1 


35.5 


46.5 


36.3 


60 


50.3 


32.7 


49.7 


33.6 


49.1 


34.4 


48.5 


35.3 


47.9 


36.1 


47.3 


36.9 


61 


51.2 


33.2 


50.6 


34.1 


50.0 


35.0 


49.4 


35.9 


48.7 


36.7 


48.1 


37.6 


62_ 


52.0 


33.8 


51.4 


34.7 


50.8 


35.6 


50.2 


36.4 


49.5 


37.3 


48.9 


38.2 


63 


52.8 


34.3 


52.2 


S5.2 


51.6 


36.1 


51.0 


37.0 


50.3 


37.9 


49.6 


38.8 


64 


53.7 


34.9 


53.1 


35.8 


52.4 


36.7 


51.8 


37.6 


51.1 


38.5 


50.4 


39.4 


65 


54.5 


35.4 


53.9 


36.3 


53.2 


37.3 


52.6 


38.2 


51.9 


39.1 


51.2 


40.0 


66 


55.4 


35.9 


54.7 


36.9 


54.1 


37.9 


53.4 


38.8 


52.7 


39.7 


52.0 


40.6 


67 


56.2 


36.5 


55.5 


37.5 


54.9 


38.4 


54.2 


39.4 


53.5 


40.3 


52.8 


41.2 


68 


57.0 


37.0 


56.4 


38.0 


55.7 


39.0 


55.0 


40.0 


54.3 


40.9 


53.6 


41.9 


69 


57.9 


37.6 


57.2 


38.6 


56.5 


39.6 


55.8 


40.6 


55.1 


41.5 


54.4 


42.5 


70 


58.7 


38.1 


58.0 


39.1 


57.3 


40.2 


56.6 


41.1 


55.9 


42.1 


55.2 


43.1 


71 


59.5 


38.7 


58.9 


39.7 


58.2 


40.7 


57.4 


41.7 


56.7 


42.7 


55.9 


43.7 


72 


60.4 


39.2 


59.7 


40.3 


59.0 


41.3 


58.2 


42.3 


57.5 


43.3 


56.7 


44.3 


73 


61.2 


39.8 


60.5 


40.8 


59.8 


41.9 


59.1 


42.9 


58.3 


43.9 


57.5 


44.9 


74 


62.1 


40.3 


61.3 


41.4 


60.6 


42.4 


59.9 


43.5 


59.1 


44.5 


58.3 


45.6 


75 


62.9 


40.8 


62.2 


41.9 


61.4 


43.0 


60.7 


44.1 


59.9 


45.1 


59.1 


46.2 


76 


63.7 


41.4 


63.0 


42.5 


62.3 


43.6 


61.5 


44.7 


60.7 


45.7 


59.9 


46.8 


77 


64.6 


41.9 


63.8 


43.1 


63.1 


44.2 


62.3 


45.3 


61.5 


46.3 


60.7 


47.4 


78 


65.4 


42.5 


64.7 


43.6 


63.9 


44.7 


63.1 


45.8 


62.3 


46.9 


61.5 


48.0 


79 


66.3 


43.0 


65.5 


44.2 


64.7 


45.3 


63.9 


46.4 


63.1 


47.5 


62.3 


48.6 


80 


67.1 


43.6 


66.3 


44.7 


65.5 


45.9 


64.7 


47.0 


63.9 


48.1 


63.0 


49.3 


81 


67.9 


44.1 


67.2 


45.3 


66.4 


46.5 


65.5 


47.6 


64.7 


48.7 


63.8 


49.9 


82 


68.8 


44.7 


68.0 


45.9 


67.2 


47.0 


66.3 


48.2 


65.5 


49.3 


64.6 


50.5 


83 


69.6 


45.2 


68.8 


46.4 


68.0 


47.6 


67.1 


48.8 


66.3 


50.0 


65.4 


51.1 


84 


70.4 


45.7 


69.6 


47.0 


68.8 


48.2 


68.0 


49.4 


67.1 


50.6 


66.2 


51.7 


85 


71.3 


46.3 


70.5 


47.5 


69.6 


48.8 


68.8 


50.0 


67.9 


51.2 


67.0 


52.3 


86 


72.1 


46.8 


71.3 


48.1 


70.4 


49.3 


69.6 


50.5 


68.7 


51.8 


67.8 


52.9 


87 


73.0 


47.4 


72.1 


48.6 


71.3 


49.9 


70.4 


51.1 


69.5 


52.4 


68.6 


53.6 


88 


73.8 


47.9 


73.0 


49.2 


72.1 


50.5 


71.2 


51.7 


70.3 


53.0 


69.3 


54.2 


89 


74.6 


48.5 


73.8 


49.8 


72.9 


51.0 


72.0 


52.3 


71.1 


53.6 


70.1 


54.8 


90 


75.5 


49.0 


74.6 


50.3 


73.7 


51.6 


72.8 


52.9 


71.9 


54.2 


70.9 


55.4 


91 


76.3 


49.6 


75.4 


50.9 


74.5 


52.2 


73.6 


53.5 


72.7 


54.8 


71.7 


56.0 


92 


77.2 


50.1 


76.3 


51.4 


75.4 


52.8 


74.4 


54.1 


73.5 


55.4 


72.5 


56.6 


93 


78.0 


50.7 


77.1 


52.0 


76.2 


53.3 


75.2 


54.7 


74.3 


56.0 


73.3 


57.3 


94 


78.8 


51.2 


77.9 


52.6 


77.0 


53.9 


76.0 


55.3 


75.1 


56.6 


74.1 


57.9 


95 


79.7 


51.7 


78.8 


53.1 


77.8 


54.5 


76.9 


55.8 


75.9 


57.2 


74.9 


58.5 


96 


80.5 


52.3 


79.6 


53.7 


78.6 


55.1 


77.7 


56.4 


76.7 


57.8 


75.6 


59.1 


97 


81.4 


52.8 


80.4 


54.2 


79.5 


55.6 


78.5 


57.0 


77.5 


58.4 


76.4 


59.7 


98 


82.2 


53.4 


81.2 


54.8 


80.3 


56.2 


79.3 


57.6 


78.3 


59.0 


77.2 


60.3 


99 


83.0 


53.9 


82.1 


55.4 


81.1 


56.8 


80.1 


58.2 


79.1 


59.6 


78.0 


61.0 


100 


83.9 


54.5 


82.9 


55.9 


81.9 


57.4 


80.9 


58.8 


79.9 


60.2 


78.8 


61.6 


600 


503.2 


326.8 


497.4 


335.5 


491.5 


344.1 


485.4 


352.7 


479.2 


361.1 


472.8 


369.4 


700 


587.0 


381.3 


580.3 


391.4 


573.5 


401.5 


566.2 


411.4 


559.0 


421.3 


551.6 


430.8 


800 


671.0 


435.7 


663.3 


447.4 


655.4 


458.8 


647.3 


470.2 


638.9 


481.5 


630.4 


492.5 


900 


754.8 


490.1 


746.1 


503.2 


737.2 


516.2 


728.1 


528.9 


718.6 


541.7 


709.1 


554.0 




Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 




(123, 237, 


(124, 236, 


(125, 235, 


(126, 234, 


(127, 233, 


(128, 232. 




303) 


304") 


305) 


306) 


307) 


308) 




57 


5 Pt. 56 


55 . 


54 


4J Pt. 53 


52 



The 5-Pt. or 56 Courses are : N.E. by E., S.E. by E., N.W. by W., S.W. by W. 



166 



Table 1. Traverse Table 





3| Pt. 39 


40 


41 


3f Pt. 42 


43 


44 


4 Pt. 45 




(141, 219, 


(140, 220, 


(139, 221, 


(138, 222, 


(137, 223, 


(136, 224, 


(135, 225, 


DlST. 


321) 


320) 


319) 


318) 


317) 


316) 


315) 




Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


1 


0.8 


0.6 


0.8 


0.6 


0.8 


0.7 


0.7 


0.7 


0.7 


0.7 


0.7 


0.7 


0.7 


0.7 


2 


1.6 


1.3 


1.5 


1.3 


1.5 


1.3 


1.5 


1.3 


1.5 


1.4 


1.4 


1.4 


1.4 


1.4 


3 


2.3 


1.9 


2.3 


1.9 


2.3 


2.0 


2.2 


2.0 


2.2 


2.0 


2.2 


2.1 


2.1 


2.1 


4 


3.1 


2.5 


3.1 


2.6 


3.0 


2.6 


3.0 


2.7 


2.9 


2.7 


2.9 


2.8 


2.8 


2.8 


5 


3.9 


3.1 


3.8 


3.2 


3.8 


3.3 


3.7 


3.3 


3.7 


3.4 


3.6 


3.5 


3.5 


3.5 


6 


4.7 


3.8 


4.6 


3.9 


4.5 


3.9 


4.5 


4.0 


4.4 


4.1 


4.3 


4.2 


4.2 


4.2 


7 


5.4 


4.4 


5.4 


4.5 


5.3 


4.6 


5.2 


4.7 


5.1 


4.8 


5.0 


4.9 


4.9 


4.9 


8 


6.2 


5.0 


6.1 


5.1 


6.0 


5.2 


5.9 


5.4 


5.9 


5.5 


5.8 


5.6 


5.7 


5.7 


9 


7.0 


5.7 


6.9 


5.8 


6.8 


5.9 


6.7 


6.0 


6.6 


6.1 


6.5 


6.3 


6.4 


6.4 


10 


7.8 


6.3 


7.7 


6.4 


7.5 


6.6 


7.4 


6.7 


7.3 


6.8 


7.2 


6.9 


7.1 


7.1 


11 


8.5 


6.9 


8.4 


7.1 


8.3 


7.2 


8.2 


7.4 


8.0 


7.5 


7.9 


7.6 


7.8 


7.8 


12 


9.3 


7.6 


9.2 


7.7 


9.1 


7.9 


8.9 


8.0 


8.8 


8.2 


8.6 


8.3 


8.5 


8.5 


13 


10.1 


8.2 


10.0 


8.4 


9.8 


8.5 


9.7 


8.7 


9.5 


8.9 


9.4 


9.0 


9.2 


9.2 


14 


10.9 


8.8 


10.7 


9.0 


10.6 


9.2 


10.4 


9.4 


10.2 


9.5 


10.1 


9.7 


9.9 


9.9 


15 


11.7 


9.4 


11.5 


g.e 


11.3 


9.8 


11.1 


10.0 


11.0 


10.2 


10.8 


10.4 


10.6 


10.6 


16 


12.4 


10.1 


12.3 


10.3 


12.1 


10.5 


11.9 


10.7 


11.7 


10.9 


11.5 


11.1 


11.3 


11.3 


17 


13.2 


10.7 


13.0 


10.9 


12.8 


11.2 


12.6 


11.4 


12.4 


11.6 


12.2 


11.8 


12.0 


12.0 


18 


14.0 


11.3 


13.8 


11.6 


13.6 


11.8 


13.4 


12.0 


13.2 


12.3 


12.9 


12.5 


12.7 


12.7 


19 


14.8 


12.0 


14.6 


12.2 


14.3 


12.5 


14.1 


12.7 


13.9 


13.0 


13.7 


13.2 


13.4 


13.4 


20 


15.5 


12.6 


15.3 


12.9 


15.1 


13.1 


14.9 


13.4 


14.6 


13.6 


14.4 


13.9 


14.1 


14.1 


21 


16.3 


13.2 


16.1 


13.5 


15.8 


13.8 


15.6 


14.1 


15.4 


14.3 


15.1 


14.6 


14.8 


14.8 


22 


17.1 


13.8 


16.9 


14.1 


16.6 


14.4 


16.3 


14.7 


16.1 


15.0 


15.8 


15.3 


15.6 


15.6 


23 


17.9 


14.5 


17.6 


14.8 


17.4 


15.1 


17.1 


15.4 


16.8 


15.7 


16.5 


16.0 


16.3 


16.3 


24 


18.7 


15.1 


18.4 


15.4 


18.1 


15.7 


17.8 


16.1 


17.6 


16.4 


17.3 


16.7 


17.0 


17.0 


25 


19.4 


15.7 


19.2 


16.1 


18.9 


16.4 


18.6 


16.7 


18.3 


17.0 


18.0 


17.4 


17.7 


17.7 


26 


20.2 


16.4 


19.9 


16.7 


19.6 


17.1 


19.3 


17.4 


190 


17.7 


18.7 


18.1 


18.4 


18.4' 


27 


21.0 


17.0 


20.7 


17.4 


20.4 


17.7 


20.1 


18.1 


19.7 


18.4 


19.4 


18.8 


19.1 


19.1 


28 


21.8 


17.6 


21.4 


18.0 


21.1 


18.4 


20.8 


18.7 


20.5 


19.1 


20.1 


19.5 


19.8 


19.8 


29 


22.5 


18.3 


22.2 


18.6 


21.9 


19.0 


21.6 


19.4 


21.2 


19.8 


20.9 


20.1 


20.5 


20.5 


30 


23.3 


18.9 


23.0 


19.3 


22.6 


19.7 


22.3 


20.1 


21.9 


20.5 


21.6 


20.8 


21.2 


21.2 


31 


24.1 


19.5 


23.7 


19.9 


23.4 


20.3 


23.0 


20.7 


22.7 


21.1 


22.3 


21.5 


21.9 


21.9 


32 


24.9 


20.1 


24.5 


20.6 


24.2 


21.0 


23.8 


21.4 


23.4 


21.8 


23.0 


22.2 


22.6 


22.6 


33 


25.6 


20.8 


25.3 


21.2 


24.9 


21.6 


24.5 


22.1 


24.1 


22.5 


23.7 


22.9 


23.3 


23.3 


34 


26.4 


21.4 


26.0 


21.9 


25.7 


22.3 


25.3 


22.8 


24.9 


23.2 


24.5 


23.6 


24.0 


24.0 


35 


27.2 


22.0 


26.8 


22.5 


26.4 


23.0 


26.0 


23.4 


25.6 


23.9 


25.2 


24.3 


24.7 


24.7 


36 


28.0 


22.7 


27.6 


23.1 


27.2 


23.6 


26.8 


24.1 


26.3 


24.6 


25.9 


25.0 


25.5 


25.5 


37 


28.8 


23.3 


28.3 


23.8 


27.9 


24.3 


27.5 


24.8 


27.1 


25.2 


26.6 


25.7 


26.2 


26.2 


38 


29.5 


23.9 


29.1 


24.4 


28.7 


24.9 


28.2 


25.4 


27.8 


25.9 


27.3 


26.4 


26.9 


26.9 


39 


30.3 


24.5 


29.9 


25.1 


29.4 


25.6 


29.0 


26.1 


28.5 


26.6 


28.1 


27.1 


27.6 


27.6 


40 


31.1 


25.2 


30.6 


25.7 


30.2 


26.2 


29.7 


26.8 


29.3 


27.3 


28.8 


27.8 


28.3 


28.3 


41 


31.9 


25.8 


31.4 


26.4 


30.9 


26.9 


30.5 


27.4 


30.0 


28.0 


29.5 


28.5 


29.0 


29.0 


42 


32.6 


26.4 


32.2 


27.0 


31.7 


27.6 


31.2 


28.1 


30.7 


28.6 


30.2 


29.2 


29.7 


29.7 


43 


33.4 


27.1 


32.9 


27.6 


32.5 


28.2 


32.0 


28.8 


31.4 


29.3 


30.9 


29.9 


30.4 


30.4 


44 


34.2 


27.7 


33.7 


28.3 


33.2 


28.9 


32.7 


29.4 


32.2 


30.0 


31.7 


30.6 


31.1 


31.1 


45 


35.0 


28.3 


34.5 


28.9 


34.0 


29.5 


33.4 


30.1 


32.9 


30.7 


32.4 


31.3 


31.8 


31.8 


46 


35.7 


28.9 


35.2 


29.6 


34.7 


30.2 


34.2 


30.8 


33.6 


31.4 


33.1 


32.0 


32.5 


32.5 


47 


36.5 


29.6 


36.0 


30.2 


35.5 


30.8 


34.9 


31.4 


34.4 


32.1 


33.8 


32.6 


33.2 


33.2 


48 


37.3 


30.2 


36.8 


30.9 


36.2 


31.5 


35.7 


32.1 


35.1 


32.7 


34.5 


33.3 


33.9 


33.9 


49 


38.1 


30.8 


37.5 


31.5 


37.0 


32.1 


36.4 


32.8 


35.8 


33.4 


35.2 


34.0 


34.6 


34.6 


50 


38.9 


31.5 


38.3 


32.1 


37.7 


32.8 


37.2 


33.5 


36.6 


34.1 


36.0 


34.7 


35.4 


35.4 


100 


77.7 


62.9 


76.6 


64.3 


75.5 


65.6 


74.3 


66.9 


73.1 


68.2 


71.9 


69.5 


70.7 


70.7 


200 


155.4 


125.9 


153.2 


128.6 


150.9 


131.2 


148.6 


133.8 


146.3 


136.4 


143.9 


138.9 


141.4 


141.4 


300 


233.1 


188.8 


229.8 


192.8 


226.4 


196.8 


222.9 


200.7 


219.4 


204.6 


215.8 


208.4 


212.1 


212.1 


400 


310.9 


251.7 


306.4 


257.1 


301.9 


262.4 


297.3 


267.7 


292.6 


272.8 


287.7 


277.9 


282.8 


282.8 


500 


388.6 


314.7 


383.0 


321.4 


377.3 


328.0 


371.6 


334.6 


365.7 


341.0 


359.7 


347.3 


353.5 


353.5 




Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 




(129, 231, 


(130, 230, 


(131, 229, 


(132, 228, 


(133, 227, 


(134, 226, 


(135, 225, 




309) 


310) 


311 


312) 


313) 


314) 


315) 




4| Pt. 51 


50 


49 


41 Pt. 48 


47 


46 


4 Pt. 45 



The 4-Pt. or 45 Courses are : N.E., N.W., S.E., S.W. 



Table 1. Traverse Table 



167 





3^ Pt, 39 


40 


41 


3f Pt. 42 


43 


44 


4 Pt. 45 




(141, 219, 


(140, 220, 


(139, 221, 


(138, 222, 


(137, 223, 


(136, 224, 


(135, 225, 


DlST. 


321) 


320) 


319) 


318) 


- 317) 


316) 


315) 




Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


51 


39.6 


32.1 


39.1 


32.8 


38.5 


33.5 


37.9 


34.1 


37.3 


34.8 


36.7 


35.4 


36.1 


36.1 


52 


40.4 


32.7 


39.8 


33.4 


39.2 


34.1 


38.6 


34.8 


38.0 


35.5 


37.4 


36.1 


36.8 


36.8 


53 


41.2 


33.4 


40.6 


34.1 


40.0 


34.8 


39.4 


35.5 


38.8 


36.1 


38.1 


36.8 


37.5 


37.5 


54 


42.0 


34.0 


41.4 


34.7 


40.8 


35.4 


40.1 


36.1 


39.5 


36.8 


38.8 


37.5 


38.2 


38.2 


55 


42.7 


34.6 


42.1 


35.4 


41.5 


36.1 


40.9 


36.8 


40.2 


37.5 


39.6 


38.2 


38.9 


38.9 


56 


43.5 


35.2 


42.9 


36.0 


42.3 


36.7 


41.6 


37.5 


41.0 


38.2 


40.3 


38.9 


39.6 


39.6 


57 


44.3 


35.9 


43.7 


36.6 


43.0 


37.4 


42.4 


38.1 


41.7 


38.9 


41.0 


39.6 


40.3 


40.3 


58 


45.1 


36.5 


44.4 


37.3 


43.8 


38.1 


43.1 


38.8 


42.4 


39.6 


41.7 


40.3 


41.0 


41.0 


59 


45.9 


37.1 


45.2 


37.9 


44.5 


38.7 


43.8 


39.5 


43.1 


40.2 


42.4 


41.0 


41.7 


41.7 


60 


46.6 


37.8 


46.0 


38.6 


45.3 


39.4 


44.6 


40.1 


43.9 


40.9 


43.2 


41.7 


42.4 


42.4 


61 


47.4 


38.4 


46.7 


39.2 


46.0 


40.0 


45.3 


40.8 


44.6 


41.6 


43.9 


42.4 


43.1 


43.1 


62 


48.2 


39,0 


4L 


39.9 


46.8 


40.7 


46.1 


41".5 


45.3 


42.3 


44.6 


43.1 


43.8 


43.8 


V63 


49.0 


39.6 


18.3 


40.5 


47.5 


41.3 


46.8 


42.2 


46.1 


43.0 


45.3 


43.8 


44.5 


44.5 


64 


49.7 


40.3 


49!0 


41.1 


48.3 


42.0 


47.6 


42.8 


46.8 


43.6 


46.0 


44.5 


45.3 


45.3 


65 


50.5 


40.9 


49.8 


41.8 


49.1 


42.6 


48.3 


43.5 


47.5 


44.3 


46.8 


45.2 


46.0 


46.0 


66 


51.3 


41.5 


50.6 


42.4 


49.8 


43.3 


49.0 


44.2 


48.3 


45.0 


47.5 


45.8 


46.7 


46.7 


67 


52.1 


42.2 


51.3 


43.1 


50.6 


44.0 


49.8 


44.8 


49.0 


45.7 


48.2 


46.5 


47.4 


47.4 


68 


52.8 


42.8 


52.1 


43.7 


51.3 


44.6 


50.5 


45.5 


49.7 


46.4 


48.9 


47.2 


48.1 


48.1 


69 


53.6 


43.4 


52.9 


44.4 


52.1 


45.3 


51.3 


46.2 


50.5 


47.1 


49.6 


47.9 


48.8 


48.8 


70 


54.4 


44.1 


53.6 


45.0 


52.8 


45.9 


52.0 


46.8 


51.2 


47.7 


50.4 


48.6 


49.5 


49.5 


71 


55.2 


44.7 


54.4 


45.6 


53.6 


46.6 


52.8 


47.5 


51.9 


48.4 


51.1 


49.3 


50.2 


50.2 


72 


56.0 


45.3 


55.2 


46.3 


54.3 


47.2 


53.5 


48.2 


52.7 


49.1 


51.8 


50.0 


50.9 


50.9 


73 


56.7 


45.9 


55.9 


46.9 


55.1 


47.9 


54.2 


48.8 


53.4 


49.8 


52.5 


50.7 


51.6 


51.6 


74 


57.5 


46.6 


56.7 


47.6 


55.8 


48.5 


55.0 


49.5 


54.1 


50.5 


53.2 


51.4 


52.3 


52.3 


75 


58.3 


47.2 


57.5 


48.2 


56.6 


49.2 


55.7 


50.2 


54.9 


51.1 


54.0 


52.1 


53.0 


53.0 


76 


59.1 


47.8 


58.2 


48.9 


57.4 


49.9 


56.5 


50.9 


55.6 


51.8 


54.7 


52.8 


53.7 


53.7 


77 


59.8 


48.5 


59.0 


49.5 


58.1 


50.5 


57.2 


51.5 


56.3 


52.5 


55.4 


53.5 


54.4 


54.4 


78 


60.6 


49.1 


59.8 


50.1 


58.9 


51.2 


58.0 


52.2 


57.0 


53.2 


56.1 


54.2 


55.2 


55.2 


79 


61.4 


49.7 


60.5 


50.8 


59.6 


51.8 


58.7 


52.9 


57.8 


53.9 


56.8 


54.9 


55.9 


55.9 


80 


62.2 


50.3 


61.3 


51.4 


60.4 


52.5 


59.5 


53.5 


58.5 


54.6 


57.5 


55.6 


56.6 


56.6 


81 


62.9 


51.0 


62.0 


52.1 


61.1 


53.1 


60.2 


54.2 


59.2 


55.2 


58.3 


56.3 


57.3 


57.3 


82 


63.7 


51.6 


62.8 


52.7 


61.9 


53.8 


60.9 


54.9 


60.0 


55.9 


59.0 


57.0 


58.0 


58.0 


83 


64.5 


52.2 


63.6 


53.4 


62.6 


54.5 


61.7 


55.5 


60.7 


56.6 


59.7 


57.7 


58.7 


58.7 


84 


65.3 


52.9 


64.3 


54.0 


63.4 


55.1 


62.4 


56.2 


61.4 


57.3 


60.4 


58.4 


59.4 


59.4 


85 


66.1 


53.5 


65.1 


54.6 


64.2 


55.8 


63.2 


56.9 


62.2 


58.0 


61.1 


59.0 


60.1 


60.1 


86 


66.8 


54.1 


65.9 


55.3 


64.9 


56.4 


63.9 


57.5 


62.9 


58.7 


61.9 


59.7 


60.8 


60.8 


87 


67.6 


54.8 


66.6 


55.9 


65.7 


57.1 


64.7 


58.2 


63.6 


59.3 


62.6 


60.4 


61.5 


61.5 


88 


68.4 


55.4 


67.4 


56.6 


66.4 


57.7 


65.4 


58.9 


64.4 


60.0 


63.3 


61.1 


62.2 


62.2 


89 


69.2 


56.0 


68.2 


57.2 


67.2 


58.4 


66.1 


59.6 


65.1 


60.7 


64.0 


61.8 


62.9 


62.9 


90 


69.9 


56.6 


68.9 


57.9 


67.9 


59.0 


66.9 


60.2 


65.8 


61.4 


64.7 


62.5 


63.6 


63.6 


91 


70.7 


57.3 


69.7 


58.5 


68.7 


59.7 


67.6 


60.9 


66.6 


62.1 


65.5 


63.2 


64.3 


64.3 


92 


71.5 


57.9 


70.5 


59.1 


69.4 


60.4 


68.4 


61.6 


67.3 


62.7 


66.2 


63.9 


65.1 


65.1 


93 


72.3 


58.5 


71.2 


59.8 


70.2 


61.0 


69.1 


62.2 


68.0 


63.4 


66.9 


64.6 


65.8 


65.8 


94 


73.1 


59.2 


72.0 


60.4 


70.9 


61.7 


69.9 


62.9 


68.7 


64.1 


67.6 


65.3 


66.5 


66.5 


95 


73.8 


59.8 


72.8 


61.1 


71.7 


62.3 


70.6 


63.6 


69.5 


64.8 


68.3 


66.0 


67.2 


67.2 


96 


74.6 


60.4 


73.5 


61.7 


72.5 


63.0 


71.3 


64.2 


70.2 


65.5 


69.1 


66.7 


67.9 


67.9 


97 


75.4 


61.0 


74.3 


62.4 


73.2 


63.6 


72.1 


64.9 


70.9 


66.2 


69.8 


67.4 


68.6 


68.6 


98 


76.2 


61.7 


75.1 


63.0 


74.0 


64.3 


72.8 


65.6 


71.7 


66.8 


70.5 


68.1 


69.3 


69.3 


99 


76.9 


62.3 


75.8 


63.6 


74.7 


64.9 


73.6 


66.2 


72.4 


67.5 


71.2 


68.8 


70.0 


70.0 


100 


77.7 


62.9 


76.6 


64.3 


75.5 


65.6 


74.3 


66.9 


73.1 


68.2 


71.9 


69.5 


70.7 


70.7 


600 


466.3 


377.6 


459.6 


385.7 


452.8 


393.6 


445.9 


401.5 


438.8 


409.2 


431.6 


416.8 


424.3 


424.3 


700 


543.9 


440.6 


536.3 


450.0 


528.3 


459.2 


520.2 


468.4 


511.9 


477.4 


503.5 


486.3 


495.0 


495.0 


800 


621.8 


503.5 


613.0 


514.2 


603.9 


524.8 


594.6 


535.3 


585.1 


545.6 


575.4 


555.8 


565.7 


565.7 


900 


699.3 


566.3 


689.5 


578.5 


679.2 


590.3 


668.8 


602.2 


658.2 


613.8 


647.3 


625.2 


636.3 


636.3 




Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 




(129, 231, 


(130, 230, 


(131, 229, 


(132, 228, 


(133, 227, 


(134, 226, 


(135, 225, 




309) 


310) 


311) 


312) 


313) 


314) 


315) 




4| Pt. 51 


50 


49 


41 Pt. 48 


47 


46 


4 Pt. 45 



The 4-Pt. or 45 Courses are : N.E., N.W., S.E., S.W. 



168 



Table 2 



To CHANGE LONG. DIFF. INTO DEP., SUBTRACT TABULAR NUMBER 
FROM LONG. DIFF. 



LONG. 
DIFP. 


MIDDLE LATITUDE 


OR 

DEP. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


11 


12 


13 


14 


15 


1 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


2 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.1 


0.1 


0.1 


3 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.1 


0.1 


0.1 


0.1 


0.1 


4 


00 


00 


00 


0.0 


00 


00 


00 


0.0 


00 


0.1 


01 


01 


0.1 


0.1 


1 


5 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.1 


0.1 


0.1 


0.1 


0.1 


0.1 


0.2 


6 


00 


00 


0.0 


00 


00 


00 


00 


1 


1 


0.1 


01 


1 


0? 


0.2 


02 


7 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.1 


0.1 


0.1 


0.1 


0.1 


0.2 


0.2 


0.2 


0.2 


8 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.1 


0.1 


0.1 


0.1 


0.1 


0.2 


0.2 


0.2 


0.3 


9 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.1 


0.1 


0.1 


0.1 


0.2 


0.2 


0.2 


0.3 


0.3 


10 


0.0 


0.0 


0.0 


0.0 


0.0 


0.1 


0.1 


0.1 


0.1 


0.2 


0.2 


0.2 


0.3 


0.3 


0.3 


11 


0.0 


0.0 


0.0 


0.0 


0.0 


0.1 


0.1 


0.1 


0.1 


0.2 


0.2 


0.2 


0.3 


0.3 


0.4 


12 


0.0 


0.0 


0.0 


0.0 


0.0 


0.1 


0.1 


0.1 


0.1 


0.2 


0.2 


0.3 


0.3 


0.4 


0.4 


13 


0.0 


0.0 


0.0 


0.0 


0.0 


0.1 


0.1 


0.1 


0.2 


0.2 


0.2 


0.3 


0.3 


0.4 


0.4 


14 


0.0 


0.0 


0.0 


0.0 


0.1 


0.1 


0.1 


0.1 


0.2 


0.2 


0.3 


0.3 


0.4 


0.4 


0.5 


15 


0.0 


0.0 


0.0 


0.0 


0.1 


0.1 


0.1 


0.1 


0.2 


0.2 


0.3 


0.3 


0.4 


0.4 


0.5 


16 


0.0 


0.0 


0.0 


0.0 


0.1 


0.1 


0.1 


0.2 


0.2 


0.2 


0.3 


0.3 


0.4 


0.5 


0.5 


17 


0.0 


0.0 


0.0 


0.0 


0.1 


0.1 


0.1 


0.2 


0.2 


0.3 


0.3 


0.4 


0.4 


0.5 


0.6 


18 


0.0 


0.0 


0.0 


0.0 


0.1 


0.1 


0.1 


0.2 


0.2 


0.3 


0.3 


0.4 


0.5 


0.5 


0.6 


19 


0.0 


0.0 


0.0 


0.0 


0.1 


0.1 


0.1 


0.2 


0.2 


0.3 


0.3 


0.4 


0.5 


0.6 


0.6 


20 


0.0 


0.0 


0.0 


0.0 


0.1 


0.1 


0.1 


0.2 


0.2 


0.3 


0.4 


0.4 


0.5 


0.6 


0.7 


21 


0.0 


0.0 


0.0 


0.1 


0.1 


0.1 


0.2 


0.2 


0.3 


0.3 


0.4 


0.5 


0.5 


0.6 


0.7 


22 


0.0 


0.0 


0.0 


0.1 


0.1 


0.1 


0.2 


0.2 


0.3 


0.3 


0.4 


0.5 


0.6 


0.7 


0.7 


23 


0.0 


0.0 


0.0 


0.1 


0.1 


0.1 


0.2 


0.2 


0.3 


0.3 


0.4 


0.5 


0.6 


0.7 


0.8 


24 


0.0 


0.0 


0.0 


0.1 


0.1 


0.1 


0.2 


0.2 


0.3 


0.4 


0.4 


0.5 


0.6 


0.7 


0.8 


25 


0.0 


0.0 


0.0 


0.1 


0.1 


0.1 


0.2 


0.2 


0.3 


0.4 


0.5 


0.5 


0.6 


0.7 


0.9 


26 


0.0 


0.0 


0.0 


0.1 


0.1 


0.1 


0.2 


0.3 


0.3 


0.4 


0.5 


0.6 


0.7 


0.8 


0.9 


27 


0.0 


0.0 


0.0 


0.1 


0.1 


0.1 


0.2 


0.3 


0.3 


0.4 


0.5 


0.6 


0.7 


0.8 


0.9 


28 


0.0 


0.0 


0.0 


0.1 


0.1 


0.2 


0.2 


0.3 


0.3 


0.4 


0.5 


0.6 


0.7 


0.8 


1.0 


29 


0.0 


0.0 


0.0 


0.1 


0.1 


0.2 


0.2 


0.3 


0.4 


0.4 


0.5 


0.6 


0.7 


0.9 


1.0 


30 


0.0 


0.0 


0.0 


0.1 


0.1 


0.2 


0.2 


0.3 


0.4 


0.5 


0.6 


0.7 


0.8 


0.9 


1.0 


31 


0.0 


0.0 


0.0 


0.1 


0.1 


0.2 


0.2 


0.3 


0.4 


0.5 


0.6 


0.7 


0.8 


0.9 


1.1 


32 


0.0 


0.0 


0.0 


0.1 


0.1 


0.2 


0.2 


0.3 


0.4 


0.5 


0.6 


0.7 


0.8 


1.0 


1.1 


33 


0.0 


0.0 


0.0 


0.1 


0.1 


0.2 


0.2 


0.3 


0.4 


0.5 


0.6 


0.7 


0.8 


1.0 


1.1 


34 


0.0 


0.0 


0.0 


0.1 


0.1 


0.2 


0.3 


0.3 


0.4 


0.5 


0.6 


0.7 


0.9 


1.0 


1.2 


35 


0.0 


0.0 


0.0 


0.1 


0.1 


0.2 


0.3 


0.3 


0.4 


0.5 


0.6 


0.8 


0.9 


1.0 


1.2 


36 


0.0 


0.0 


0.0 


0.1 


0.1 


0.2 


0.3 


0.4 


0.4 


0.5 


0.7 


0.8 


0.9 


1.1 


1.2 


37 


0.0 


0.0 


0.1 


0.1 


0.1 


0.2 


0.3 


0.4 


0,5 


0.6 


0.7 


0.8 


0.9 


1.1 


1.3 


38 


0.0 


0.0 


0.1 


0.1 


0.1 


0.2 


0.3 


0.4 


0.5 


0.6 


0.7 


0.8 


1.0 


1.1 


1.3 


39 


0.0 


0.0 


0.1 


0.1 


0.1 


0.2 


0.3 


0.4 


0.5 


0.6 


0.7 


0.9 


1.0 


1.2 


1.3 


40 


0.0 


0.0 


0.1 


0.1 


0.2 


0.2 


0.3 


0.4 


0.5 


0.6 


0.7 


0.9 


1.0 


1.2 


1.4 


41 


0.0 


0.0 


0.1 


0.1 


0.2 


0.2 


0.3 


0.4 


0.5 


0.6 


0.8 


0.9 


1.1 


1.2 


1.4 


42 


0.0 


0.0 


0.1 


0.1 


0.2 


0.2 


0.3 


0.4 


0.5 


0.6 


0.8 


0.9 


1.1 


1.2 


1.4 


43 


0.0 


0.0 


0.1 


0.1 


0.2 


0.2 


0.3 


0.4 


0.5 


0.7 


0.8 


0.9 


1.1 


1.3 


1.5 


44 


0.0 


0.0 


0.1 


0.1 


0.2 


0.2 


0.3 


0.4 


0.5 


0.7 


0.8 


1.0 


1.1 


1.3 


1.5 


45 


0.0 


0.0 


0.1 


0.1 


0.2 


0.2 


0.3 


0.4 


0.6 


0.7 


0.8 


1.0 


1.2 


1.3 


1.5 


46 


0.0 


0.0 


0.1 


0.1 


0.2 


0.3 


0.3 


0.4 


0.6 


0.7 


0.8 


1.0 


1.2 


1.4 


1.6 


47 


0.0 


0.0 


0.1 


0.1 


0.2 


0.3 


0.4 


0.5 


0.6 


0.7 


0.9 


1.0 


1.2 


1.4 


1.6 


48 


0.0 


0.0 


0.1 


0.1 


0.2 


0.3 


0.4 


0.5 


0.6 


0.7 


0.9 


1.0 


1.2 


1.4 


1.6 


49 


0.0 


0.0 


0.1 


0.1 


0.2 


0.3 


0.4 


0.5 


0.6 


0.7 


0.9 


1.1 


1.3 


1.5 


1.7 


50 


0.0 


0.0 


0.1 


0.1 


0.2 


0.3 


0.4 


0.5 


0.6 


0.8 


0.9 


1.1 


1.3 


1.5 


1.7 


100 


0.0 


0.1 


0.1 


0.2 


0.4 


0.5 


0.7 


1.0 


1.2 


1.5 


1.8 


2.2 


2.6 


3.0 


3.4 


200 


0.0 


0.1 


0.3 


0.5 


0.8 


1.1 


1.5 


1.9 


2.5 


3.0 


3.7 


4.4 


5.1 


5.9 


6.8 


300 


0.0 


0.2 


0.4 


0.7 


1.1 


1.6 


2.2 


2.9 


3.7 


4.6 


5.5 


6.6 


7.7 


8.9 


10.2 


400 


0.1 


0.2 


0.6 


1.0 


1.5 


2.2 


3.0 


3.9 


4.9 


6.1 


7.4 


8.7 


10.2 


11.9 


13.7 


500 


0.1 


0.3 


0.7 


1.2 


1.9 


2.7 


3.7 


4.9 


6.2 


7.6 


9.2 


10.9 


12.8 


14.9 


17.0 




1.00 


1.00 


1.00 


1.00 


1.00 


1.01 


1.01 


1.01 


1.01 


1.02 


1.02 


1.02 


1.03 


1.03 


1.04 




FACTOB 



To CHANGE DEP. INTO LONG. DIFF., MULTIPLY TABULAR NUMBER BY 
FACTOR AT FOOT OF COLUMN, AND ADD PRODUCT TO DEP. 



Table 2 



169 



To CHANGE LONG. DIFF. INTO DEP. SUBTRACT TABULAR NUMBER 
FROM LONG. DIFF. 



LONG. 

DIFF. 


MIDDLE LATITUDE 


OR 

DEP. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


11 


12 


13 


14 


15 


51 


0.0 


0.0 


0.1 


0.1 


0.2 


0.3 


0.4 


0.5 


0.6 


0.8 


0.9 


1.1 


1.3 


1.5 


1.7 


52 


0.0 


0.0 


0.1 


0.1 


0.2 


0.3 


0.4 


0.5 


0.6 


0.8 


1.0 


1.1 


1.3 


1.5 


1.8 


53 


0.0 


0.0 


0.1 


0.1 


0.2 


0.3 


0.4 


0.5 


0.7 


0.8 


1.0 


1.2 


1.4 


1.6 


1.8 


54 


0.0 


0.0 


0.1 


0.1 


0.2 


0.3 


0.4 


0.5 


0.7 


0.8 


1.0 


1.2 


1.4 


1.6 


1.8 


55 


0.0 


0.0 


0.1 


0.1 


0.2 


0.3 


0.4 


0.5 


0.7 


0.8 


1.0 


1.2 


1.4 


1.6 


1.9 


56 


0.0 


0.0 


0.1 


0.1 


0.2 


0.3 


0.4 


0.5 


0.7 


0.9 


1.0 


1.2 


1.4 


1.7 


1.9 


57 


0.0 


0.0 


0.1 


0.1 


0.2 


0.3 


0.4 


0.6 


0.7 


0.9 


1.0 


1.2 


1.5 


1.7 


1.9 


58 


0.0 


0.0 


0.1 


0.1 


0.2 


0.3 


0.4 


0.6 


0.7 


0.9 


1.1 


1.3 


1.5 


1.7 


2.0 


59 


0.0 


0.0 


0.1 


0.1 


0.2 


0.3 


0.4 


0.6 


0.7 


0.9 


1.1 


1.3 


1.5 


1.8 


2.0 


60 


0.0 


0.0 


0.1 


0.1 


0.2 


0.3 


0.4 


0.6 


0.7 


0.9 


1.1 


1.3 


1.5 


1.8 


2.0 


61 


0.0 


0.0 


0.1 


0.1 


0.2 


0.3 


0.5 


0.6 


0.8 


0.9 


1.1 


1.3 


1.6 


1.8 


2.1 


62 


0.0 


0.0 


0.1 


0.2 


0.2 


0.3 


0.5 


0.6 


0.8 


0.9 


1.1 


1.4 


1.6 


1.8 


2.1 


63 


0.0 


0.0 


0.1 


0.2 


0.2 


0.3 


0.5 


0.6 


0.8 


1.0 


1.2 


1.4 


1.6 


1.9 


2.1 


64 


0.0 


0.0 


0.1 


0.2 


0.2 


0.4 


0.5 


0.6 


0.8 


1.0 


1.2 


1.4 


1.6 


1.9 


2.2 


65 


0.0 


0.0 


0.1 


0.2 


0.2 


0.4 


0.5 


0.6 


0.8 


1.0 


1.2 


1.4 


1.7 


1.9 


2.2 


66 


0.0 


0.0 


0.1 


0.2 


0.3 


0.4 


0.5 


0.6 


0.8 


1.0 


1.2 


1.4 


1.7 


2.0 


2.2 


67 


0.0 


0.0 


0.1 


0.2 


0.3 


0.4 


0.5 


0.7 


0.8 


1.0 


1.2 


1.5 


1.7 


2.0 


2.3 


68 


0.0 


0.0 


0.1 


0.2 


0.3 


0.4 


0.5 


0.7 


0.8 


1.0 


1.2 


1.5 


1.7 


2.0 


2.3 


69 


0.0 


0.0 


0.1 


0.2 


0.3 


0.4 


0.5 


0.7 


0.8 


1.0 


1.3 


1.5 


1.8 


2.0 


2.4 


70 


0.0 


0.0 


0.1 


0.2 


0.3 


0.4 


0.5 


0.7 


0.9 


1.1 


1.3 


1.5 


1.8 


2.1 


2.4 


71 


0.0 


0.0 


0.1 


0.2 


0.3 


0.4 


0.5 


0.7 


0.9 


1.1 


1.3 


1.6 


1.8 


2.1 


2.4 


72 


0.0 


0.0 


0.1 


0.2 


0.3 


0.4 


0.5 


0.7 


0.9 


1.1 


1.3 


1.6 


1.8 


2.1 


2.5 


73 


0.0 


0.0 


0.1 


0.2 


0.3 


0.4 


0.5 


0.7 


0.9 


1.1 


1.3 


1.6 


1.9 


2.2 


2.5 


74 


0.0 


0.0 


0.1 


0.2 


0.3 


0.4 


0.6 


0.7 


0.9 


1.1 


1.4 


1.6 


1.9 


2.2 


2.5 


75 


0.0 


0.0 


0.1 


0.2 


0.3 


0.4 


0.6 


0.7 


0.9 


1.1 


1.4 


1.6 


1.9 


2.2 


2.6 


76 


0.0 


0.0 


0.1 


0.2 


0.3 


0.4 


0.6 


0.7 


0.9 


1.2 


1.4 


1.7 


1.9 


2.3 


2.6 


77 


0.0 


0.0 


0.1 


0.2 


0.3 


0.4 


0.6 


0.7 


0.9' 


1.2 


1.4 


1.7 


2.0 


2.3 


2.6 


78 


0.0 


0.0 


0.1 


0.2 


0.3 


0.4 


0.6 


0.8 


1.0 


1.2 


1.4 


1.7 


2.0 


2.3 


2.7 


79 


0.0 


0.0 


0.1 


0.2 


0.3 


0.4 


0.6 


0.8 


1.0 


1.2 


1.5 


1.7 


2.0 


2.3 


2.7 


80 


0.0 


0.0 


0.1 


0.2 


0.3 


0.4 


0.6 


0.8 


1.0 


1.2 


1.5 


1.7 


2.1 


2.4 


2.7 


81 


0.0 


0.0 


0.1 


0.2 


0.3 


0.4 


0.6 


0.8 


1.0 


1.2 


1.5 


1.8 


2.1 


2.4 


2.8 


82 


0.0 


0.0 


0.1 


0.2 


0.3 


0.4 


0.6 


0.8 


1.0 


1.2 


1.5 


1.8 


2.1 


2.4 


2.8 


83 


0.0 


0.1 


0.1 


0.2 


0.3 


0.5 


0.6 


0.8 


1.0 


1.3 


1.5 


1.8 


2.1 


2.5 


2.8 


84 


0.0 


0.1 


0.1 


0.2 


0.3 


0.5 


0.6 


0.8 


1.0 


1.3 


1.5 


1.8 


2.2 


2.5 


2.9 


85 


0.0 


0.1 


0.1 


0.2 


0.3 


0.5 


0.6 


0.8 


1.0 


1.3 


1.6 


1.9 


2.2 


2.5 


2.9 


86 


0.0 


0.1 


0.1 


0.2 


0.3 


0.5 


0.6 


0.8 


1.1 


1.3 


1.6 


1.9 


2.2 


2.6 


2.9 


87 


0.0 


0.1 


0.1 


0.2 


0.3 


0.5 


0.6 


0.8 


1.1 


1.3 


1.6 


1.9 


2.2 


2.6 


3.0 


88 


0.0 


0.1 


0.1 


0.2 


0.3 


0.5 


0.7 


0.9 


1.1 


1.3 


1.6 


1.9 


2.3 


2.6 


3.0 


89 


0.0 


0.1 


0.1 


0.2 


0.3 


0.5 


0.7 


0.9 


1.1 


1.4 


1.6 


1.9 


2.3 


2.6 


3.0 


90 


0.0 


0.1 


0.1 


0.2 


0.3 


0.5 


0.7 


0.9 


1.1 


1.4 


1.7 


2.0 


2.3 


2.7 


3.1 


91 


0.0 


0.1 


0.1 


0.2 


0.3 


0.5 


0.7 


0.9 


1.1 


1.4 


1.7 


2.0 


2.3 


2.7 


3.1 


92 


0.0 


0.1 


0.1 


0.2 


0.4 


0.5 


0.7 


0.9 


1.1 


1.4 


1.7 


2.0 


2.4 


2.7 


3.1 


93 


0.0 


0.1 


0.1 


0.2 


0.4 


0.5 


0.7 


0.9 


1.1 


1.4 


1.7 


2.0 


2.4 


2.8 


3.2 


94 


0.0 


0.1 


0.1 


0.2 


0.4 


0.5 


0.7 


0.9 


1.2 


1.4 


1.7 


2.1 


2.4 


2.8 


3.2 


95 


0.0 


0.1 


0.1 


0.2 


0.4 


0.5 


0.7 


0.9 


1.2 


1.4 


1.7 


2.1 


2.4 


2.8 


3.2 


96 


0.0 


0.1 


0.1 


0.2 


0.4 


0.5 


0.7 


0.9 


1.2 


1.5 


1.8 


2.1 


2.5 


2.9 


3.3 


97 


0.0 


0.1 


0.1 


0.2 


0.4 


0.5 


0.7 


0.9 


1.2 


1.5 


1.8 


2.1 


2.5 


2.9 


3.3 


98 


0.0 


0.1 


0.1 


0.2 


0.4 


0.5 


0.7 


1.0 


1.2 


1.5 


1.8 


2.1 


2.5 


2.9 


3.3 


99 


0.0 


0.1 


0.1 


0.2 


0.4 


0.5 


0.7 


1.0, 


1.2 


1.5 


1.8 


2.2 


2.5 


2.9 


3.4 


100 


0.0 


0.1 


0.1 


0.2 


0.4 


0.5 


0.7 


1.0 


1.2 


1.5 


1.8 


2.2 


2.6 


3.0 


3.4 


600 


0.1 


0.4 


0.8 


1.4 


2.3 


3.3 


4.5 


5.8 


7.4 


9.1 


10.0 


13.1 


15.4 


17.8 


20.5 


700 


0.2 


0.5 


1.0 


1.8 


2.8 


3.9 


5.1 


6.7 


8.7 


10.5 


12.9 


15.3 


17.9 


20.8 


23.9 


800 


0.2 


0.5 


1.1 


2.0 


3.1 


4.4 


5.9 


7.7 


9.8 


12.1 


14.8 


17.5 


20.6 


23.8 


27.3 


900 


0.3 


0.7 


1.4 


2.4 


3.6 


5.0 


6.7 


8.7 


11.2 


13.7 


16.7 


19.8 


23.2 


26.8 


30.8 




1.00 


1.00 


1.00 


1.00 


1.00 


1.01 


1.01 


1.01 


1.01 


1.02 


1.02 


1.02 


1.03 


1.03 


1.04 




FACTOR 



To CHANGE DEP. INTO LONG. DIFF. MULTIPLY TABULAR NUMBER BY 
FACTOR AT FOOT OF COLUMN AND ADD PRODUCT TO DEP. 



170 



Table 2 



To CHANGE LONG. DIFF. INTO DEP., SUBTRACT TABULAR NUMBER 
FROM LONG. DIFF. 



LONG. 
DIFF. 


MIDDLE LATITUDE 


DEP. 


16 


17 


18 


19 


20 


21 


22 


23 


24 


25 


26 


27 


28 


1 


0.0 


0.0 


0.0 


0.1 


0.1 


0.1 


0.1 


0.1 


0.1 


0.1 


0.1 


0.1 


0.1 


2 


0.1 


0.1 


0.1 


0.1 


0.1 


0.1 


0.1 


0.2 


0.2 


0.2 


0.2 


0.2 


0.2 


3 


0.1 


0.1 


0.1 


0.2 


0.2 


0.2 


0.2 


0.2 


0.3 


0.3 


0.3 


0.3 


0.4 


4 


0.2 


0.2 


0.2 


0.2 


0.2 


0.3 


0.3 


0.3 


0.3 


0.4 


0.4 


0.4 


0.5 


5 


0.2 


0.2 


0.2 


0.3 


0.3 


0.3 


0.4 


0.4 


0.4 


0.5 


0.5 


0.5 


0.6 


6 


0.2 


0.3 


0.3 


0.3 


0.4 


0.4 


0.4 


0.5 


0.5 


0.6 


0.6 


0.7 


0.7 


7 


0.3 


0.3 


0.3 


0.4 


0.4 


0.5 


0.5 


0.6 


0.6 


0.7 


0.7 


0.8 


0.8 


8 


0.3 


0.3 


0.4 


0.4 


0.5 


0.5 


0.6 


0.6 


0.7 


0.7 


0.8 


0.9 


0.9 


9 


0.3 


0.4 


0.4 


0.5 


0.5 


0.6 


0.7 


0.7 


0.8 


0.8 


0.9 


1.0 


1.1 


10 


0.4 


0.4 


0.5 


0.5 


0.6 


0.7 


0.7 


0.8 


0.9 


0.9 


1.0 


1.1 


1.2 


11 


0.4 


0.5 


0.5 ' 


0.6 


0.7 


0.7 


0.8 


0.9 


1.0 


1.0 


1.1 


1.2 


1.3 


12 


05 


0.5 


0.6 


0.7 


0.7 


0.8 


0.9 


1.0 


1.0 


1.1 


1.2 


1.3 


1 4 


13 


0.5 


0.6 


0.6 


0.7 


0.8 


0.9 


0.9 


1.0 


1.1 


1.2 


1.3 


1.4 


1.5 


14 


0.5 


0.6 


0.7 


0.8 


0.8 


0.9 


1.0 


1.1 


1.2 


1.3 


1.4 


1.5 


1.6 


15 


0.6 


0.7 


0.7 


0.8 


0.9 


1.0 


1.1 


1.2 


1.3 


1.4 


1.5 


1.6 


1.8 


16 


0.6 


0.7 


0.8 


0.9 


1.0 


1.1 


1.2 


1.3 


1.4 


1.5 


1.6 


1.7 


1.9 


17 


0.7 


0.7 


0.8 


0.9 


1.0 


1.1 


1.2 


1.4 


1.5 


1.6 


1.7 


1.9 


2.0 


18 


0.7 


0.8 


0.9 


1.0 


1.1 


1.2 


1.3 


1.4 


1.6 


1.7 


1.8 


2.0 


2.1 


19 


0.7 


0.8 


0.9 


1.0 


1.1 


1.3 


1.4 


1.5 


1.6 


1.8 


1.9 


2.1 


2.2 


20 


0.8 


0.9 


1.0 


1.1 


1.2 


1.3 


1.5 


1.6 


1.7 


1.9 


2.0 


2.2 


2.3 


21 


0.8 


0.9 


1.0 


1.1 


1.3 


1.4 


1.5 


1.7 


1.8 


2.0 


2.1 


2.3 


2.5 


22 


0.9 


1.0 


1.1 


1.2 


1.3 


1.5 


1.6 


1.7 


1.9 


2.1 


2.2 


2.4 


2.6 


23 


0.9 


1.0 


1.1 


1.3 


1.4 


1.5 


1.7 


1.8 


2.0 


2.2 


2.3 


2.5 


2.7 


24 


0.9 


1.0 


1.2 


1.3 


1.4 


1.6 


1.7 


1.9 


2.1 


2.2 


2.4 


2.6 


2.8 


25 


1.0 


1.1 


1.2 


1.4 


1.5 


1.7 


1.8 


2.0 


2.2 


2.3 


2.5 


2.7 


2.9 


26 


1.0 


1.1 


1.3 


1.4 


1.6 


1.7 


1.9 


2.1 


2.2 


2.4 


2.6 


2.8 


3.0 


27 


1.0 


1.2 


1.3 


1.5 


1.6 


1.8 


2.0 


2.1 


2.3 


2.5 


2.7 


2.9 


3.2 


28 


1.1 


1.2 


1.4 


1.5 


1.7 


1.9 


2.0 


2.2 


2.4 


2.6 


2.8 


3.1 


3.3 


29 


1.1 


1.3 


1.4 


1.6 


1.7 


1.9 


2.1 


2.3 


2.5 


2.7 


2.9 


3.2 


3.4 


30 


1.2 


1.3 


1.5 


1.6 


1.8 


2.0 


2.2 


2.4 


2.6 


2-.8 


3.0 


3.3 


3.5 


31 


1.2 


1.4 


1.5 


1.7 


1.9 


2.1 


2.3 


2.5 


2.7 


2.9 


3.1 


3.4 


3.6 


32 


1.2 


1.4 


1.6 


1.7 


1.9 


2.1 


2.3 


2.5 


2.8 


3.0 


3.2 


3.5 


3.7 


33 


1.3 


1.4 


1.6 


1.8 


2.0 


2.2 


2.4 


2.6 


2.9 


3.1 


3.3 


3.6 


3.9 


34 


1.3 


1.5 


1.7 


1.9 


2.1 


2.3 


2.5 


2.7 


2.9 


3.2 


3.4 


3.7 


4.0 


35 


1.4 


1.5 


1.7 


1.9 


2.1 


2.3 


2.5 


2.8 


3.0 


3.3 


3.5 


3.8 


4.1 


36 


1.4 


1.6 


1.8 


2.0 


2.2 


2.4 


2.6 


2.9 


3.1 


3.4 


3.6 


3.9 


4.2 


37 


1.4 


1.6 


1.8 


2.0 


2.2 


2.5 


2.7 


2.9 


3.2 


3.5 


3.7 


4.0 


4.3 


38 


1.5 


1.7 


1.9 


2.1 


2.3 


2.5 


2.8 


3.0 


3.3 


3.6 


3.8 


4.1 


4.4 


39 


1.5 


1.7 


1.9 


2.1 


2.4 


2.6 


2.8 


3.1 


3.4 


3.7 


3.9 


4.3 


4.6 


40 


1.5 


1.7 


2.0 


2.2 


2.4 


2.7 


2.9 


3.2 


3.5 


3.7 


4.0 


4.4 


4.7 


41 


1.6 


1.8 


2.0 


2.2 


2.5 


2.7 


3.0 


3.3 


3.5 


3.8 


4.1 


4.5 


4.8 


42 


1.6 


1.8 


2.1 


2.3 


2.5 


2.8 


3.1 


3.3 


3.6 


3.9 


4.3 


4.6 


4.9 


43 


1.7 


1.9 


2.1 


2.3 


2.6 


2.9 


3.1 


3.4 


3.7 


4.0 


1.4 


4.7 


5.0 


44 


1.7 


1.9 


2.2 


2.4 


2.7 


2.9 


3.2 


3.5 


3.8 


4.1 


4.5 


4.8 


5.2 


45 


1.7 


2.0 


2.2 


2.5 


2.7 


3.0 


3.3 


3.6 


3.9 


4.2 


4.6 


4.9 


5.3 


46 


1.8 


2.0 


2.3 


2.5 


2.8 


3.1 


3.3 


3.7 


4.0 


4.3 


4.7 


5.0 


5.4 


47 


1.8 


2.1 


2.3 


2.6 


2.8 


3.1 


3.4 


3.7 


4.1 


4.4 


4.8 


5.1 


5.5 


48 


1.9 


2.1 


2.3 


2.6 


2.9 


3.2 


3.5 


3.8 


4.1 


4.5 


4.9 


5.2 


5.6 


49 


1.9 


2.1 


2.4 


2.7 


3.0 


3.3 


3.6 


3.9 


4.2 


4.6 


5.0 


5.3 


5.7 


50 


1.9 


2.2 


2.4 


2.7 


3.0 


3.3' 


3.6 


4.0 


4.3 


4.7 


5.1 


5.4 


5.9 


100 


3.9 


4.4 


4.9 


5.4 


6.0 


6.6 


7.3 


7.9 


8.6 


9.4 


10.1 


10.9 


11.7 


200 


7.7 


8.7 


9.8 


10.9 


12.1 


13.3 


14.6 


15.9 


17.3 


18.7 


20.2 


21.8 


23.4 


300 


11.6 


13.1 


14.7 


16.3 


18.1 


19.9 


21.8 


23.8 


25.9 


28.1 


30.4 


32.7 


35.1 


400 


15.5 


17.5 


19.6 


21.8 


24.1 


26.6 


29.1 


31.8 


34.6 


37.5 


40.5 


43.6 


46.9 


500 


19.4 


21.9 


24.5 


27.2 


30.1 


33.2 


36.4 


39.8 


43.2 


46.9 


50.6 


54.5 


58.5 




1.04 


1.05 


1.05 


1.06 


1.06 


1.07 


1.08 


1.09 


1.09 


1.10 


1.11 


1.12 


1.13 




FACTOR 



To CHANGE DEP. INTO LONG. DIFF., MULTIPLY TABULAR NUMBER BY 

T^ATTOR AT TTnnT nw (^nT.rnwivr Aiun Ann PRnnrrrT TO DF.P. 



Table 2 



171 



To CHANGE LONG. DIFF. INTO DEP. SUBTRACT TABULAR NUMBER 
FROM LONG. DIFF. 



LONG. 
DIPP. 


MIDDLE LATITUDE 


OR 

DEP. 


16 


17 


18 


19 


20 


21 


22 


23 


24 


25 


26 


27 


28 


51 


2.0 


2.2 


2.5 


2.8 


3.1 


3.4 


3.7 


4.1 


4.4 


4.8 


5.2 


5.6 


6.0 


52 


2.0 


2.3 


2.5 


2.8 


3.1 


3.5 


3.8 


4.1 


4.5 


4.9 


5.3 


5.7 


6.1 


53 


2.1 


2.3 


2.6 


2.9 


3.2 


3.5 


3.9 


4.2 


4.6 


5.0 


5.4 


5.8 


6.2 


54 


2.1 


2.4 


2.6 


2.9 


3.3 


3.6 


3.9 


4.3 


4.7 


5.1 


5.5 


5.9 


6.3 


55 


2.1 


2.4 


2.7 


3.0 


3.3 


3.7 


4.0 


4.4 


4.8 


5.2 


'5.6 


6.0 


6.4 


56 


2.2 


2.4 


2.7 


3.1 


3.4 


3.7 


4.1 


4.5 


4.8 


5.2 


5.7 


6.1 


6.6 


57 


2.2 


2.5 


2.8 


3.1 


3.4 


3.8 


4.2 


4.5 


4.9 


5.3 


5.8 


6.2 


6.7 


58 


2.2 


2.5 


2.8 


3.2 


3.5 


3.9 


4.2 


4.6 


5.0 


5.4 


5.9 


6.3 


6.8 


59 


2.3 


2.6 


2.9 


3.2 


3.6 


3.9 


4.3 


4.7 


5.1 


5.5 


6.0 


6.4 


6.9 


60 


2.3 


2.6 


2.9 


3.3 


3.6 


4.0 


4.4 


4.8 


5.2 


5.6 


6.1 


6.5 


7.0 


61 


2.4 


2.7 


3.0 


3.3 


3.7 


4.1 


4.4 


4.8 


5.3 


5.7 


6.2 


6.6 


7.1 


62 


2.4 


2.7 


3.0 


3.4 


3.7 


4.1 


4.5 


4.9 


5.4 


5.8 


6.3 


6.8 


7.3 


63 


2.4 


2.8 


3.1 


3.4 


3.8 


4.2 


4.6 


5.0 


5.4 


5.9 


6.4 


6.9 


7.4 


64 


2.5 


2.8 


3.1 


3.5 


3.9 


4.3 


4.7 


5.1 


5.5 


6.0 


6.5 


7.0 


7.5 


65 


2.5 


2.8 


3.2 


3.5 


3.9 


4.3 


4.7 


5.2 


5.6 


6.1 


6.6 


7.1 


7.6 


66 


2.6 


2.9 


3.2 


3.6 


4.0 


4.4 


4.8 


5.2 


5.7 


6.2 


6.7 


7.2 


7.7 


67 


2.6 


2.9 


3.3 


3.7 


4.0 


4.5 


4.9 


5.3 


5.8 


6.3 


6.8 


7.3 


7.8 


68 


2.6 


3.0 


3.3 


3.7 


4.1 


4.5 


5.0 


5.4 


5.9 


6.4 


6.9 


7.4 


8.0 


69 


2.7 


3.0 


3.4 


3.8 


4.2 


4.6 


5.0 


5.5 


6.0 


6.5 


7.0 


7.5 


8.1 


70 


2.7 


3.1 


3.4 


3.8 


4.2 


4.6 


5.1 


5.6 


6.1 


6.6 


7.1 


7.6 


8.2 


71 


2.8 


3.1 


3.5 


3.9 


4.3 


4.7 


5.2 


5.6 


6.1 


6.7 


7.2 


7.7 


8.3 


72 


2.8 


3.1 


3.5 


3.9 


4.3 


4.8 


5.2 


5.7 


6.2 


6.7 


7.3 


7.8 


8.4 


73 


2.8 


3.2 


3.6 


4.0 


4.4 


4.8 


5.3 


5.8 


6.3 


6.8 


7.4 


8.0 


8.5 


74 


2.9 


3.2 


3.6 


4.0 


4.5 


4.9 


5.4 


5.9 


6.4 


6.9 


7.5 


8.1 


8.7 


75 


2.9 


3.3 


3.7 


4.1 


4.5 


5.0 


5.5 


6.0 


6.5 


7.0 


7.6 


8.2 


8.8 


76 


2.9 


3.3 


3.7 


4.1 


4.6 


5.0 


5.5 


6.0 


6.6 


7.1 


7.7 


8.3 


8.9 


77 


3.0 


3.4 


3.8 


4.2 


4.6 


5.1 


5.6 


6.1 


6.7 


7.2 


7.8 


8.4 


9.0 


78 


3.0 


3.4 


3.8 


4.2 


4.7 


5.2 


5.7 


6.2 


6.7 


7.3 


7.9 


8.5 


9.1 


79 


3.1 


3.5 


3.9 


4.3 


4.8 


5.2 


5.8 


6.3 


6.8 


7.4 


8.0 


8.6 


9.2 


80 


3.1 


3.5 


3.9 


4.4 


4.8 


5.3 


5.8 


6.4 


6.9 


7.5 


8.1 


8.7 


9.4 


81 


3.1 


3.5 


4.0 


4.4 


4.9 


5.4 


5.9 


6.4 


7.0 


7.6 


8.2 


8.8 


9.5 


82 


3.2 


3.6 


4.0 


4.5 


4.9 


5.4 


6.0 


6.5 


7.1 


7.7 


8.3 


8.9 


9.6 


83 


3.2 


3.6 


4.1 


4.5 


5.0 


5.5 


6.0 


6.6 


7.2 


7.8 


8.4 


9.0 


9.7 


84 


3.3 


3.7 


4.1 


4.6 


5.1 


5.6 


6.1 


6.7 


7.3 


7.9 


8.5 


9.2 


9.8 


85 


3.3 


3.7 


4.2 


4.6 


5.1 


5.6 


6.2 


6.8 


7.3 


8.0 


8.6 


9.3 


9.9 


86 


3.3 


3.8 


4.2 


4.7 


5.2 


5.7 


6.3 


6.8 


7.4 


8.1 


8.7 


9.4 


10.1 


87 


3.4 


3.8 


4.3 


4.7 


5.2 


5.8 


6.3 


6.9 


7.5 


8.2 


8.8 


9.5 


10.2 


88 


3.4 


3.8 


4.3 


4.8 


5.3 


5.8 


6.4 


7.0 


7.6 


8.2 


8.9 


9.6 


10.3 


89 


3.4 


3.9 


4.4 


4.8 


5.4 


5.9 


6.5 


7.1 


7.7 


8.3 


9.0 


9.7 


10.4 


90 


3.5 


3.9 


4.4 


4.9 


5.4 


6.0 


6.6 


7.2 


7.8 


8.4 


9.1 


9.8 


10.5 


91 


3.5 


4.0 


4.5 


5.0 


5.5 


6.0 


6.6 


7.2 


7.9 


8.5 


9.2 


9.9 


10.7 


92 


3.6 


4.0 


4.5 


5.0 


5.5 


6.1 


6.7 


7.3 


8.0 


8.6 


9.3 


10.0 


10.8 


93 


3.6 


4.1 


4.6 


5.1 


5.6 


6.2 


6.8 


7.4 


8.0 


8.7 


9.4 


10.1 


10.9 


94 


3.6 


4.1 


4.6 


5.1 


5.7 


6.2 


6.8 


7.5 


8.1 


8.8 


9.5 


10.2 


11.0 


95 


3.7 


4.2 


4.6 


5.2 


5.7 


6.3 


6.9 


7.6 


8.2 


8.9 


9.6 


10.4 


11.1 


96 


3.7 


4.2 


4.7 


5.2 


5.8 


6.4 


7.0 


7.6 


8.3 


9.0 


9.7 


10.5 


11.2 


97 


3.8 


4.2 


4.7 


5.3 


5.8 


6.4 


7.1 


7.7 


8.4 


9.1 


9.8 


10.6 


11.4 


98 


3.8 


4.3 


4.8 


5.3 


5.9 


6.5 


7.1 


7.8 


8.5 


9.2 


9.9 


10.7 


11.5 


99 


3.8 


4.3 


4.8 


5.4 


6.0 


6.6 


7.2 


7.9 


8.6 


9.3 


10.0 


10.8 


11.6 


100 


3.9 


4.4 


4.9 


5.4 


6.0 


6.6 


7.3 


7.9 


8.6 


9.4 


10.1 


10.9 


11.7 


600 


23.2 


26.2 


29.4 


32.7 


36.2 


39.9 


43.7 


47.7 


51.9 


56.2 


60.7 


65.4 


70.2 


700 


27.2 


30.6 


34.2 


38.1 


42.1 


46.4 


50.9 


55.7 


60.5 


65.5 


70.8 


76.3 


82.0 


800 


31.0 


35.0 


39.2 


43.5 


48.2 


53.1 


58.2 


63.6 


69.2 


74.9 


80.9 


87.1 


93.7 


900 


35.0 


39.4 


44.1 


49.1 


54.3 


59.7 


65.5 


71.7 


77.9 


84.4 


91.1 


98.1 


105.5 




1.04 


1.05 


1.05 


1.06 


1.06 


1.07 


1.08 


1.09 


1.10 


1.10 


1.11 


1.12 


1.13 




FACTOB 



To CHANGE DEP. INTO LONG. DIFF. MULTIPLY TABULAR NUMBER BY 
FACTOR AT FOOT OF COLUMN, AND ADD PRODUCT TO DEP. 



172 



Table 2 



To CHANGE LONG. DIFP. INTO DEP., SUBTRACT TABULAR NUMBER 
FROM LONG. DIFF. 



LONO. 
DIFF. 


MIDDLE LATITUDE 


OR 

DEP. 


29 


30 


31 


32 


33 


34 


35 


36 


37 


38 


39 


40 


1 


0.1 


0.1 


0.1 


0.2 


0.2 


0.2 


0.2 


0.2 


0.2 


0.2 


0.2 


0.2 


2 


0.3 


0.3 


0.3 


0.3 


0.3 


0.3 


0.4 


0.4 


0.4 


0.4 


0.4 


0.5 


3 


0.4 


0.4 


0.4 


0.5 


0.5 


0.5 


0.5 


0.6 


0.6 


0.6 


0.7 


0.7 


4 


0.5 


0.5 


0.6 


0.6 


0.6 


0.7 


0.7 


0.8 


0.8 


0.8 


0.9 


0.9 


5 


0.6 


0.7 


0.7 


0.8 


0.8 


0.9 


0.9 


1.0 


1.0 


1.1 


1.1 


1.2 


6 


0.8 


0.8 


0.9 


0.9 


1.0 


1.0 


1.1 


1.1 


1.2 


1.3 


1.3 


1.4 


7 


0.9 


0.9 


1.0 


1.1 


1.1 


1.2 


1.3 


1.3 


1.4 


1.5 


1.6 


1.6 


8 


1.0 


1.1 


1.1 


1.2 


1.3 


1.4 


1.4 


1.5 


1.6 


1.7 


1.8 


1.9 


9 


1.1 


1.2 


1.3 


1.4 


1.5 


1.5 


1.6 


1.7 


1.8 


1.9 


2.0 


2.1 


10 


1.3 


1.3 


1.4 


1.5 


1.6 


1.7 


1.8 


1.9 


2.0 


2.1 


2.2 


2.3 


11 


1.4 


1.5 


1.6 


1.7 


1.8 


1.9 


2.0 


2.1 


2.2 


2.3 


2.5 


2.6 


12 


1.5 


1.6 


1.7 


1.8 


1.9 


2.1 


2.2 


2.3 


. 2.4 


2.5 


2.7 


2.8 


13 


1.6 


1.7 


1.9 


2.0 


2.1 


2.2 


2.4 


2.5 


2.6 


2.8 


2.9 


3.0 


14 


1.8 


1.9 


2.0 


2.1 


2.3 


2.4 


2.5 


2.7 


2.8 


3.0 


3.1 


3.3 


15 


1.9 


2.0 


2.1 


2.3 


2.4 


2.6 


2.7 


2.9 


3.0 


3.2 


3.3 


3.5 


16 


2.0 


2.1 


2.3 


2.4 


2.6 


2.7 


2.9 


3.1 


3.2. 


3.4 


3.6 


3.7 


17 


2.1 


2.3 


2.4 


2.6 


2.7 


2.9 


3.1 


3.2 


3.4 


3.6 


3.8 


4.0 


18 


2.3 


2.4 


2.6 


2.7 


2.9 


3.1 


3.3 


3.4 


3.6 


3.8 


4.0 


4.2 


19 


2.4 


2.5 


2.7 


2.9 


3.1 


3.2 


3.4 


3.6 


3.8 


4.0 


4.2 


4.4 


20 


2.5 


2.7 


2.9 


3.0 


3.2 


3.4 


3.6 


3.8 


4.0 


4.2 


4.5 


4.7 


21 


2.6 


2.8 


3.0 


3.2 


3.4 


3.6 


3.8 


4.0 


4.2 


4.5 


4.7 


4.9 


22 


2.8 


2.9 


3.1 


3.3 


3.5 


3.8 


4.0 


4.2 


4.4 


4.7 


4.9 


5.1 


23 


2.9 


3.1 


3.3 


3.5 


3.7 


3.9 


4.2 


4.4 


4.6 


4.9 


5.1 


5.4 


24 


3.0 


3.2 


3.4 


3.6 


3.9 


4:1 


4.3 


4.6 


4.8 


5.1 


5.3 


5.6 


25 


3.1 


3.3 


3.6 


3.8 


4.0 


4.3 


4.5 


4.8 


5.0 


5.3 


5.6 


5.8 


26 


3.3 


3.5 


3.7 


4.0 


4.2 


4.4 


4.7 


5.0 


5.2 


5.5 


5.8 


6.1 


27 


3.4 


3.6 


3.9 


4.1 


4.4 


4.6 


4.9 


5.2 


5.4 


5.7 


6.0 


6.3 


28 


3.5 


3.8 


4.0 


4.3 


4.5 


4.8 


5.1 


5.3 


5.6 


5.9 


6.2 


6.6 


29 


3.6 


3.9 


4.1 


4.4 


4.7 


5.0 


5.2 


5.5 


5.8 


6.1 


6.5 


6.8 


30 


3.8 


4.0 


4.3 


4.6 


4.8 


5.1 


5.4 


5.7 


6.0 


6.4 


6.7 


7.0 


31 


3.9 


4.2 


4.4 


4.7 


5.0 


5.3 


5.6 


5.9 


6.2 


6.6 


6.9 


7.3 


32 


4.0 


4.3 


4.6 


4.9 


5.2 


5.5 


5.8 


6.1 


6.4 


6.8 


7.1 


7.5 


33 


4.1 


4.4 


4.7 


5.0 


5.3 


5.6 


6.0 


6.3 


6.6 


7.0 


7.4 


7.7 


34 


4.3 


4.6 


4.9 


5.2 


5.5 


5.8 


6.1 


6.5 


6.8 


7.2 


7.6 


8.0 


35 


4.4 


4.7 


5.0 


5.3 


5.6 


6.0 


6.3 


6.7 


7.0 


7.4 


7.8 


8.2 


36 


4.5 


4.8 


5.1 


5.5 


5.8 


6.2 


6.5 


6.9 


7.2 


7.6 


8.0 


8.4 


37 


4.6 


5.0 


5.3 


5.6 


6.0 


6.3 


6.7 


7.1 


7.5 


7.8 


8.2 


8.7 


38 


4.8 


5.1 


5.4 


5.8 


6.1 


6.5 


6.9 


7.3 


7.7 


8.1 


8.5 


8.9 


39 


4.9 


5.2 


5.6 


5.9 


6.3 


6.7 


7.1 


7.4 


7.9 


8.3 


8.7 


9.1 


40 


5.0 


5.4 


5.7 


6.1 


6.5 


6.8 


7.2 


7.6 


8.1 


8.5 


8.9 


9.4 


41 


5.1 


5.5 


5.9 


6.2 


6.6 


7.0 


7.4 


7.8 


8.3 


8.7 


9.1 


9.6 


42 


5.3 


5.6 


6.0 


6.4 


6.8 


7.2 


7.6 


8.0 


8.5 


8.9 


9.4 


9.8 


43 


5.4 


5.8 


6.1 


6.5 


6.9 


7.4 


7.8 


8.2 


8.7 


9.1 


9.6 


10.1 


44 


5.5 


5.9 


6.3 


6.7 


7.1 


7.5 


8.0 


8.4 


8.9 


9.3 


9.8 


10.3 


45 


5.6 


6.0 


6.4 


6.8 


7.3 


7.7 


8.1 


8.6 


9.1 


9.5 


10.0 


10.5 


46 


5.8 


6.2 


6.6 


7.0 


7.4 


7.9 


8.3 


8.8 


9.3 


9.8 


10.3 


10.8 


47 


5.9 


6.3 


6.7 


7.1 


7.6 


8.0 


8.5 


9.0 


9.5 


10.0 


10.5 


11.0 


48 


6.0 


6.4 


6.9 


7.3 


7.7 


8.2 


8.7 


9.2 


9.7 


10.2 


10.7 


11.2 


49 


6.1 


6.6 


7.0 


7.4 


7.9 


8.4 


8.9 


9.4 


9.9 


10.4 


10.9 


11.5 


50 


6.3. 


6.7 


7.1 


7.6 


8.1 


8.5 


9.0 


9.5 


10.1 


10.6 


11.1 


11.7 


100 


12.5 


13.4 


14.3 


15.2 


16.1 


17.1 


18.1 


19.1 


20.1 


21.2 


22.3 


23.4 


200 


25.1 


26.8 


28.6 


30.4 


32.3 


34.2 


36.2 


38.2 


40.3 


42.4 


44.6 


46.8 


300 


37.6 


40.2 


42.9 


45.6 


48.4 


51.3 


54.3 


57.3 


60.4 


63.6 


66.9 


70.2 


400 


50.2 


53.6 


57.1 


60.8 


64.5 


68.4 


72.3 


76.4 


80.6 


84.8 


89.1 


93.6 


500 


62.7 


67.0 


71.4 


76.0 


80.7 


85.5 


90.4 


95.5 


100.7 


106.0 


111.4 


117.0 




1.14 


1.15 


1.17 


1.18 


1.19 


1.21 


1.22 


1.24 


1.25 


1.27 


1.29 


1.31 




FACTOR 



To CHANGE DEP. INTO LONG. DIFF., MULTIPLY TABULAR NUMBER BY 

" ~ 



Table 2 



173 



To CHANGE LONG. DIFF. INTO DEP. SUBTRACT TABULAR NUMBER 
FROM LONG. DIFF. 



LONG. 


MIDDLE LATITUDE 


DIFF. 




OR 

DEP. 


29 


30 


31 


32 


33 


34 


35 


36 


37 


38 


39 


40 


51 


6.4 


6.8 


7.3 


7.7 


8.2 


8.7 


9.2 


9.7 


10.3 


10.8 


ir.4 


11.9 


52 


6.5 


7.0 


7.4 


7.9 


8.4 


8.9 


9.4 


9.9 


10.5 


11.0 


11.6 


12.2 


53 


6.6 


7.1 


7.6 


8.1 


8.6 


9.1 


9.6 


10.1 


10.7 


11.2 


11.8 


12.4 


54 


6.8 


7.2 


7.7 


8.2 


8.7 


9.2 


9.8 


10.3 


10.9 


11.4 


12.0 


12.6 


55 


6.9 


7.4 


7.9 


8.4 


8.9 


9.4 


9.9 


10.5 


11.1 


11.7 


12.3 


12.9 


56 


7.0 


7.5 


8.0 


8.5 


9.0 


9.6 


10.1 


10.7 


11.3 


11.9 


12.5 


13.1 


57 


7.1 


7.6 


8.1 


8.7 


9.2 


9.7 


10.3 


10.9 


11.5 


12.1 


12.7 


13.3 


58 


7.3 


7.8 


8.3 


8.8 


9.4 


9.9 


10.5 


11.1 


11.7 


12.3 


12.9 


13.6 


59 


7.4 


7.9 


8.4 


9.0 


9.5 


10.1 


10.7 


11.3 


11.9 


12.5 


13.1 


13.8 


60 


7.5 


8.0 


8.6 


9.1 


9.7 


10.3 


10.9 


11.5 


12.1 


12.7 


13.4 


14.0 


61 


7.6 


8.2 


8.7' 


9.3 


9.8 


10.4 


11.0 


11.6 


12.3 


12.9 


13.6 


14.3 


62 


7.8 


8.3 


8.9 


9.4 


10.0 


10.6 


11.2 


11.8 


12.5 


13.1 


13.8 


14.5 


63 


7.9 


8.4 


9.0 


9.6 


10.2 


10.8 


11.4 


12.0 


12.7 


13.4 


14.0 


14.7 


64 


8.0 


8.6 


9.1 


9.7 


10.3 


10.9 


11.6 


12.2 


12.9 


13.6 


14.3 


15.0 


65 


8.1 


8.7 


9.3 


9.9 


10.5 


11.1 


11.8 


12.4 


13.1 


13.8 


14.5 


15.2 


66 


8.3 


8.8 


9.4 


10.0 


10.6 


11.3 


11.9 


12.6 


13.3 


14.0 


14.7 


15.4 


67 


8.4 


9.0 


9.6 


10.2 


10.8 


11.5 


12.1 


12.8 


13.5 


14.2 


14.9 


15.7 


68 


8.5 


9.1 


9.7 


10.3 


11.0 


11.6 


12.3 


13.0 


13.7 


14.4 


15.2 


15.9 


69 


8.7 


9.2 


9.9 


10.5 


11.1 


11.8 


12.5 


13.2 


13.9 


14.6 


15".4 


16.1 


70 


8.8 


9.4 


10.0 


10.6 


11.3 


12.0 


12.7 


13.4 


14.1 


14.8 


15.6 


16.4 


71 


8.9 


9.5 


10.1 


10.8 


11.5 


12.1 


12.8 


13.6 


14.3 


15.1 


15.8 


16.6 


72 


9.0 


9.6 


10.3 


10.9 


11.6 


12.3 


13.0 


13.8 


14.5 


15.3 


16.0 


16.8 


73 


9.2 


9.8 


10.4 


11.1 


11.8 


12.5 


13.2 


13.9 


14.7 


15.5 


16.3 


17.1 


74 


9.3 


9.9 


10.6 


11.2 


11.9 


12.7 


13.4 


14.1 


14.9 


15.7 


16.5 


17.3 


75 


9.4 


10.0 


10.7 


11.4 


12.1 


12.8 


13.6 


14.3 


15.1 


15.9 


16.7 


17.5 


76 


9.5 


10.2 


10.9 


11.5 


12.3 


13.0 


13.7 


14.5 


15.3 


16.1 


16.9 


17.8 


77 


9.7 


10.3 


11.0 


11.7 


12.4 


13.2 


13.9 


14.7 


15.5 


16.3 


17.2 


18.0 


78 


9.8 


10.5 


11.1 


11.9 


12.6 


13.3 


14.1 


14.9 


15.7 


16.5 


17.4 


18.2 


79 


9.9 


10.6 


11.3 


12.0 


12.7 


13.5 


14.3 


15.1 


15.9 


16.7 


17.6 


18.5 


80 


10.0 


10.7 


11.4 


12.2 


12.9 


13.7 


14.5 


15.3 


16.1 


17.0 


17.8 


18.7 


81 


10.2 


10.9 


11.6 


12.3 


13.1 


13.8 


14.6 


15.5 


16.3 


17.2 


18.1 


19.0 


82 


10.3 


11.0 


11.7 


12.5 


13.2 


14.0 


14.8 


15.7 


16.5 


17.4 


18.3 


19.2 


83 


10.4 


11.1 


11.9 


12.6 


13.4 


14.2 


15.0 


15.9 


16.7 


17.6 


18.5 


19.4 


84 


10.5 


11.3 


12.0 


12.8 


13.6 


14.4 


15.2 


16.0 


16.9 


17.8 


18.7 


19.7 


85 


10.7 


11.4 


12.1 


12.9 


13.7 


14.5 


15.4 


16.2 


17.1 


18.0 


18.9 


19.9 


86 


10.8 


11.5 


12.3 


13.1 


13.9 


14.7 


15.6 


16.4 


17.3 


18.2 


19.2 


20.1 


87 


10.9 


11.7 


12.4 


13.2 


14.0 


14.9 


15.7 


16.6 


17.5 


18.4 


19.4 


20.4 


88 


11.0 


11.8 


12.6 


13.4 


14.2 


15.0 


15.9 


16.8 


17.7 


18.7 


19.6 


20.6 


89 


11.2 


11.9 


12.7 


13.5 


14.4 


15.2 


16.1 


17.0 


17.9 


18.9 


19.8 


20.8 


90 


11.3 


12.1 


12.9 


13.7 


14.5 


15.4 


16.3 


17.2 


18.1 


19.1 


20.1 


21.1 


91 


11.4 


12.2 


13.0 


13.8 


14.7 


15.6 


16.5 


17.4 


18.3 


19.3 


20.3 


21.3 


92 


11.5 


12.3 


13.1 


14.0 


14.8 


15.7 


16.6 


17.6 


18.5 


19.5 


20.5 


21.5 


93 


11.7 


12.5 


13.3 


14.1 


15.0 


15.9 


16.8 


17.8 


18.7 


19.7 


20.7 


21.8 


94 


11.8 


12.6 


13.4 


14.3 


15.2 


16.1 


17.0 


18.0 


18.9 


19.9 


20.9 


22.0 


95 


11.9 


12.7 


13.6 


14.4 


15.3 


16.2 


17.2 


18.1 


19.1 


20.1 


21.2 


22.2 


96 


12.0 


12.9 


13.7 


14.6 


15:5 


16.4 


17.4 


18.3 


19.3 


20.4 


21.4 


22.5 


97 


12.2 


13.0 


13.9 


14.7 


15.6 


16.6 


17.5 


18.5 


19.5 


20.6 


21.6 


22.7 


98 


12.3 


13.1 


14.0 


14.9 


15.8 


16.8 


17.7 


18.7 


19.7 


20.8 


21.8 


22.9 


99 


12.4 


13.3 


14.1 


15.0 


16.0 


16.9 


17.9 


18.9 


19.9 


21.0 


22.1 


23.2 


100 


12.5 


13.4 


14.3 


15.2 


16.1 


17.1 


18.1 


19.1 


20.1 


21.2 


22.3 


23.4 


600 


75.2 


80.4 


85.7 


91.2 


96.8 


102.6 


108.5 


114.6 


120.8 


127.2 


133.7 


140.4 


700 


87.8 


93.9 


99.9 


106.4 


113.0 


119.7 


126.5 


133.8 


141.0 


148.4 


156.1 


163.7 


800 


100.3 


107.2 


114.2 


121.6 


129.0 


136.7 


144.6 


152.7 


161.1 


169.6 


178.2 


187.0 


900 


113.0 


120.7 


128.6 


136.8 


145.2 


153.9 


162.8 


171.9 


181.4 


190.9 


200.7 


210.5 




1.14 


1.15 


1.17 


1.18 


1.19 


1.21 


1.22 


1.24 


1.25 


1.27 


1.29 


1.31 




FACTOR 



To CHANGE DEP. INTO LONG. DIFF. MULTIPLY TABULAR NUMBER BY 

T^APTnR AT F'nn'p ni? dni.TTiww A\rr> Ann PRnrTTr"r> TT T^TT.TV 



174 



Table 2 



To CHANGE LONG. DIFP. INTO DEP., SUBTRACT TABULAR NUMBER 
FROM LONG. DIFF. 



LONG. 
DIPF. 


MIDDLE LATITUDE 


OR 
DEP. 


41 


42 


43 


44 


45; 


46 


47 


48 


49 


50 


51 


1 


0.2 


0.3 


0.3 


0.3 


0.3 


0.3 


0.3 


0.3 


0.3 


0.4 


0.4 


2 


0.5 


0.5 


0.5 


0.6 


0.6 


0.6 


0.6 


0.7 


0.7 


0.7 


0.7 


3 


0.7 


0.8 


0.8 


0.8 


0.9 


0.9 


1.0 


1.0 


1.0 


1.1 


1.1 


4 


1.0 


1.0 


1.1 


1.1 


1.2 


1.2 


1.3 


1.3 


1.4 


1.4 


15 


5 


1.2 


1.3 


1.3 


1.4 


1.5 


1.5 


1.6 


1.7 


1.7 


1.8 


1.9 


6 


1.5 


1.5 


1.6 


1.7 


1.8 


1.8 


1.9 


2.0 


2.1 


2.1 


2.2 


7 


1.7 


1.8 


1.9 


2.0 


2.1 


2.1 


2.2 


2.3 


2.4 


2.5 


2.6 


8 


2.0 


2.1 


2.1 


2.2 


2.3 


2.4 


2.5 


2.6 


2.8 


2.9 


3.0 


9 


2.2 


2.3 


2.4 


2.5 


2.6 


2.7 


2.9 


3.0 


3.1 


3.2 


3.3 


10 


2.5 


2.6 


2.7 


2.8 


2.9 


3.1 


3.2 


3.3 


3.4 


3.6 


3.7 


11 


2.7 


2.8 


3.0 


3.1 


3.2 


3.4 


3.5 


3.6 


3.8 


3.9 


4.1 


12 


2.9 


3.1 


3.2 


3.4 


3.5 


3.7 


3.8 


4.0 


4.1 


4.3 


4.4 


13 


3.2 


3.3 


3.5 


3.6 


3.8 


4.0 


4.1 


4.3 


4.5 


4.6 


4.8 


14 


3.4 


3.6 


3.8 


3.9 


4.1 


4.3 


4.5 


4.6 


4.8 


5.0 


5.2 


15 


3.7 


3.9 


4.0 


4.2 


4.4 


4.6 


4.8 


5.0 


5.2 


5.4 


5.6 


16 


3.9 


4.1 


4.3 


4.5 


4.7 


4.9 


5.1 


5.3 


5.5 


5.7 


5.9 


17 


4.2 


4.4 


4.6 


4.8 


5.0 


5.2 


5.4 


5.6 


5.8 


6.1 


6.3 


18 


4.4 


4.6 


4.8 


5.1 


5.3 


5.5 


5.7 


6.0 


6.2 


6.4 


6.7 


19 


4.7 


4.9 


5.1 


5.3 


5.6 


5.8 


6.0 


6.3 


6.5 


6.8 


7.0 


20 


4.9 


5.1 


5.4 


5.6 


5.9 


6.1 


6.4 


6.6 


6.9 


7.1 


7.4 


21 


5.2 


5.4 


5.6 


5.9 


6.2 


6.4 


6.7 


6.9 


7.2 


7.5 


7.8 


22 


5.4 


5.7 


5.9 


6.2 


6.4 


6.7 


7.0 


7.3 


7.6 


7.9 


8.2 


23 


5.6 


5.9 


6.2 


6.5 


6.7 


7.0 


7.3 


7.6 


7.9 


8.2 


8.5 


24 


5.9 


6.2 


6.4 


6.7 


7.0 


7.3 


7.6 


7.9 


8.3 


8.6 


8.9 


25 


6.1 


6.4 


6.7 


7.0 


7.3 


7.6 


8.0 


8.3 


8.6 


8.9 


9.3 


26 


64 


6.7 


7.0 


7.3 


7.6 


7.9 


8.3 


8.6 


8.9 


9.3 


96 


27 


6.6 


6.9 


7.3 


7.6 


7.9 


8.2 


8.6 


8.9 


9.3 


9.6 


10.0 


28 


6.9 


7.2 


7.5 


7.9 


8.2 


8.5 


8.9 


9.3 


9.6 


10.0 


10.4 


29 


7 1 


7.4 


7.8 


8.1 


8.5 


8.9 


9.2 


9.6 


10.0 


10.4 


107 


30 


7.4 


7.7 


8.1 


8.4 


8.8 


9.2 


9.5 


9.9 


10.3 


10.7 


11.1 


31 


7.6 


8.0 


8.3 


8.7 


9.1 


9.5 


9.9 


10.3 


10.7 


11.1 


11.5 


32 


7.8 


8.2 


8.6 


9.0 


9.4 


9.8 


10.2 


10.6 


11.0 


11.4 


11.9 


33 


8.1 


8.5 


8.9 


9.3 


9.7 


10.1 


10.5 


10.9 


11.4 


11.8 


12.2 


34 


8.3 


8.7 


9.1 


9.5 


10.0 


10.4 


10.8 


11.2 


11.7 


12.1 


12.6 


35 


8.6 


9.0 


9.4 


9.8 


10.3 


10.7 


11.1 


11.6 


12.0 


12.5 


13.0 


36 


8.8 


9.2 


9.7 


10.1 


10.5 


11.0 


11.4 


11.9 


12.4 


12.9 


13.3 


37 


9.1 


9.5 


9.9 


10.4 


10.8 


11.3 


11.8 


12.2 


12.7 


13.2 


13.7 


38 


9.3 


9.8 


10.2 


10.7 


11.1 


11.6 


12.1 


12.6 


13.1 


13.6 


14.1 


39 


9.6 


10.0 


10.5 


10.9 


11.4 


11.9 


12.4 


12.9 


13.4 


13.9 


14.5 


40 


9.8 


10.3 


10..7 


11.2 


11.7 


12.2 


12.7 


13.2 


13.8 


14.3 


14.8 


41 


10.1 


10.5 


11.0 


11.5 


12.0 


12.5 


13.0 


13.6 


14.1 


14.6 


15.2 


42 


10.3 


10.8 


11.3 


11.8 


12.3 


12.8 


13.4 


13.9 


14.4 


15.0 


15.6 


43 


10.5 


11.0 


11.6 


12.1 


12.6 


13.1 


13.7 


14.2 


14.8 


15.4 


15.9 


44 


10.8 


11.3 


11.8 


12.3 


12.9 


13.4 


14.0 


14.6 


15.1 


15.7 


16.3 


45 


11.0 


11.6 


12.1 


12.6 


13.2 


13.7 


14.3 


14.9 


15.5 


16.1 


16.7 


46 


11.3 


11.8 


12.4 


12.9 


13.5 


14.0 


14.6 


15.2 


15.8 


16.4 


17.1 


47 


11.5 


12.1 


12.6 


13.2 


13.8 


14.4 


14.9 


15.6 


16.2 


16.8 


17.4 


48 


11.8 


12.3 


12.9 


13.5 


14.1 


14.7 


15.3 


15.9 


16.5 


17.1 


17.8 


49 


12.0 


12.6 


13.2 


13.8 


14.4 


15.0 


15.6 


16.2 


16.9 


17.5 


18.2 


50 


12.3 


12.8 


13.4 


14.0 


14.6 


15.3 


15.9 


16.5 


17.2 


17.9 


18.5 


100 


24.5 


25.7 


26.9 


28.1 


29.3 


30.5 


31.8 


33.1 


34,4 


35.7 


37.1 


200 


49.1 


51.4 


53.7 


56.1 


58.6 


61.1 


63.6 


66.2 


68.8 


71.4 


74.1 


300 


73.6 


77.1 


80.6 


84.2 


87.9 


91.6 


95.4 


99.3 


103.2 


107.2 


111.2 


400 


98.1 


102.7 


107.4 


112.3 


117.2 


122.1 


127.2 


132.3 


137.6 


142.9 


148.3 


500 


122.7 


128.4 


134.3 


140.3 


146.5 


152.7 


159.0 


165.4 


172.0 


178.6 


185.3 




1.33 


1.35 


1.37 


1.39 


1.41 


1.44 


1.47 


1.50 


1.52 


1.56 


1.59 




FACTOR 



To CHANGE DEP. INTO LONG. DIFF., MULTIPLY TABULAR NUMBER BY 



Table 2 



175 



To CHANGE LONG. DIFF. INTO DEP. SUBTRACT TABULAR 'NUMBER 
FROM LONG. DIFF. 



LONG. 
DIFF. 


MIDDLE LATITUDE 


DEP. 


41 


42 


43 


44 


45 


46- 


47 


48 


49 


50 


51 


51 


12.5 


13.1 


13.7 


14.3 


14.9 


15.6 


16.2 


16.9 


17,5 


18.2 


18.9 


52 


12.8 


13.4 


14.0 


14.6 


15.2 


15.9 


16.5 


17.2 


17.9 


18.6 


19.3 


53 


13.0 


13.6 


14.2 


14.9 


15.5 


16.2 


16.9 


17.5 


18.2 


18.9 


19.6 


54 


13.2 


13.9 


14.5 


15.2 


15.8 


16.5 


17.2 


17.9 


18.6 


19.3 


20.0 


55 


13.5 


14.1 


14,8 


15.4 


16.1 


16.8 


17.5 


18.2 


18.9 


19.6 


20.4 


56 


13.7 


14.4 


lo.O 


"15.7 


16.4 


17.1 


17.8 


18.5 


19.3 


20.0 


20.8 


57 


14.0 


14.6 


15.3 


16.0 


16.7 


17.4 


18.1 


18.9 


19.6 


20.4 


21.1 


58 


14.2 


14.9 


15.6 


16.3 


17.0 


17.7 


18.4 


19.2 


19.9 


20.7 


21.5 


59 


14.5 


15.2 


15.9 


16.6 


17.3 


18.0 


18.8 


19.5 


20.3 


21.1 


21.9 


60 


14.7 


15.4 


16.1 


16.8 


17.6 


18.3 


19.1 


19.9 


20.6 


21.4 


22.2 


61 


15.0 


15.7 


16.4 


17.1 


17.9 


18.6 


19.4 


20.2 


21.0 


21.8 


22.6 


62 


15 9 


15.9 


16.7 


17.4 


18.2 


18 9 


19 7 


20 5 


21.3 


22.1 


?3 


63 


15.5 


16.2 


16.9 


17.7 


18.5 


19.2 


20.0 


20.8 


21.7 


22.5 


23.4 


64 


15.7 


16.4 


17.2 


18.0 


18.7 


19.5 


20.4 


21.2 


22.0 


22.9 


23.7 


65 


15.9 


16.7 


17.5 


18.2 


19.0 


19.8 


20.7 


21.5 


22.4 


23.2 


24.1 


66 


16.2 


17.0 


17.7 


18.5 


19.3 


20.2 


21.0 


21.8 


22.7 


23.6 


24.5 


67 


164 


17.2 


18.0 


18.8 


19.6 


20 5 


21.3 


22.2 


23.0 


23.9 


?4 8 


68 


16.7 


17.5 


18.3 


19.1 


19.9 


20.8 


21.6 


22.5 


23.4 


24.3 


25.2 


69 


16.9 


17.7 


18.5 


19.4 


20.2 


21.1 


21.9 


22.8 


23.7 


24.6 


25.6 


70 


17 ? 


18.0 


18.8 


19.6 


20.5 


21 4 


22.3 


23.2 


24.1 


25.0 


?5 9 


71 


17.4 


18.2 


19.1 


19.9 


20.8 


21.7 


22.6 


23.5 


24.4 


25.4 


26.3 


72 


177 


18.5 


19.3 


20.2 


21.1 


22 


22.9 


23.8 


24.8 


25.7 


?67 


73 


17.9 


18.8 


19.6 


20.5 


21.4 


22.3 


23.2 


24.2 


25.1 


26.1 


27.1 


74 


18.2 


19.0 


19.9 


20.8 


21.7 


22.6 


23.5 


24.5 


25.5 


26.4 


27.4 


75 


18.4 


19.3 


20.1 


21.0 


22.0 


22.9 


23.9 


24.8 


25.8 


26.8 


27.8 


76 


18.6 


19.5 


20.4 


21.3 


22.3 


23.2 


24.2 


25.1 


26.1 


27.1 


28.2 


77 


18 9 


19 8 


20 7 


21.6 


22 6 


23 5 


24.5 


25.5 


26.5 


27.5 


?85 


78 


19.1 


20.0 


21.0 


21.9 


22.8 


23.8 


24.8 


25.8 


26.8 


27.9 


28.9 


79 


19.4 


20.3 


21.2 


22.2 


23.1 


24.1 


25.1 


26.1 


27.2 


28.2 


29.3 


80 


19.6 


20.5 


21.5 


22.5 


23.4 


24.4 


25'.4 


26.5 


27.5 


28.6 


29.7 


81 


19.9 


20.8 


21.8 


22.7 


23.7 


24.7 


25.8 


26.8 


27.9 


28.9 


30.0 


82 


20.1 


21.1 


22.0 


23.0 


24.0 


25.0 


26.1 


27.1 


28.2 


29.3 


30.4 


83 


20.4 


21.3 


22.3 


23.3 


24.3 


25.3 


26.4 


27.5 


28.5 


29.6 


30.8 


84 


20.6 


21.6 


22.6 


23.6 


24.6 


25.6 


26.7 


27.8 


28.9 


30.0 


31.1 


85 


20.8 


21.8 


22.8 


23.9 


24.9 


26.0 


27.0 


28.1 


29.2 


30.4 


31.5 


86 


21.1 


22.1 


23.1 


24.1 


25.2 


26.3 


27.3 


28.5 


29.6 


30.7 


31.9 


87 


21.3 


22.3 


23.4 


24.4 


25.5 


26.6 


27.3; 


28.8 


29.9 


31.1 


32.2 


88 


21.6 


22.6 


23.6 


24.7 


25.8 


26.9 


28.0 


29.1 


30.3 


31.4 


32.6 


89 


21.8 


22.9 


23.9 


25.0 


26.1 


27.2 


28.3 


29.4 


30.6 


31.8 


33.0 


90 


22.1 


23.1 


24.2 


25.3 


26.4 


27.5 


28.6 


29.8 


31.0 


32.1 


33.4 


91 


22.3 


23.4 


24.4 


25.5 


26.7 


27.8 


28.9 


30.1 


31.3 


32.5 


33.7 


92 


22.6 


23.6 


24.7 


25.8 


26.9 


28.1 


29.3 


30.4 


31.6 


32.9 


34.1 


93 


22.8 


23.9 


25.0 


26.1 


27.2 


28.4 


29.6 


30.8 


32.0 


33.2 


34.5 


94 


23.1 


24.1 


25.3 


26.4 


27.5 


28.7 


29.9 


31.1 


32.3 


33.6 


34.8 


95 


23.3 


24.4 


25.5 


26.7 


27.8 


29.0 


30.2 


31.4 


32.7 


33.9 


35.2 


96 


23.5 


24.7 


25.8 


26.9 


28.1 


29.3 


30.5 


31.8 


33.0 


34.3 


35.6 


97 


23.8 


24.9 


26.1 


27.2 


28.4 


29.6 


30.8 


32.1 


33.4 


34.6 


36.0 


98 


24.0 


25.2 


26.3 


27.5 


28.7 


29.9 


31.2 


32.4 


33.7 


35.0 


36.3 


99 


24.3 


25.4 


26.6 


27.8 


29.0 


30.2 


31.5 


32.8 


34.1 


35.4 


36.7 


100 


24.5 


25.7 


26.9 


28.1 


29.3 


30.5 


31.8 


33.1 


34.4 


35.7 


37.1 


600 


147.2 


154.1 


161.2 


168.4 


175.7 


183.2 


190.8 


198.5 


206U 


214.3 


222.4 


700 


171.7 


179.8 


188.1 


196.5 


205.0 


213.7 


222.6 


231.6 


240.8 


250.0 


259.4 


800 


196.1 


205.4 


214.9 


224.6 


234.3 


244.2 


254.4 


264.7 


275.2 


285.8 


296.5 


900 


220.8 


231.2 


241.8 


252.7 


263.7 


274.8 


286.2 


297.8 


309.7 


321.5 


333.7 




1.33 


1.35 


1.37 


1.39 


1.41 


1.44 


1.47 


1.50 


1.52 


1.56 


1.59 




FACTOR 



To CHANGE DEP. INTO LONG. DIFF. MULTIPLY TABULAR NUMBER BY 
FACTOR AT FOOT OF COLUMN AND ADD PRODUCT TO DEP. 



176 



Table 2 



To CHANGE LONG. DIFF. INTO DEP., SUBTRACT TABULAR NUMBER 
FROM LONG. DIFF. 



LONG. 
DIPF. 

OK 


MIDDLE LATITUDE 


DBF. 


52 


53 


54 


55 


56 


57 


58 


59 


60 


1 


0.4 


0.4 


0.4 


0.4 


0.4 


0.5 


0.5 


0.5 


0.5 


2 


0.8 


0.8 


0.8 


0.9 


0.9 


0.9 


0.9 


1.0 


1.0 


3 


1.2 


1.2 


1.2 


1.3 


1.3 


1.4 


1.4 


1.5 


1.5 


4 


1.5 


1.6 


1.6 


1.7 


1.8 


1.8 


1.9 


1.9 


2.0 


5 


1.9 


2.0 


2.1 


2.1 


2.2 


2.3 


2.4 


2.4 


2.5 


6 


2.3 


2.4 


2.5 


2.6 


2.6 


2.7 


2.8 


2.9 


3.0 


7 


2.7 


2.8 


2.9 


3.0 


3.1 


3.2 


3.3 


3.4 


3.5 


8 


3.1 


3.2 


3.3 


3.4 


3.5 


3.6 


3.8 


3.9 


4.0 


9 


3.5 


3.6 


3.7 


3.8 


4.0 


4.1 


4.2 


4.4 


4.5 


10 


3.8 


4.0 


4.1 


4.3 


4.4 


4.6 


4.7 


4.8 


5.0 


11 


4.2 


4.4 


4.5 


4.7 


4.8 


5.0 


5.2 


5.3 


5.5 


12 


4.6 


4.8 


4.9 


5.1 


5.3 


5.5 


5.6 


5.8 


6.0 


13 


5.0 


5.2 


5.4 


5.5 


5.7 


5.9 


6.1 


6.3 


6.5 


14 


5.4 


5.6 


5.8 


6.0 


6.2 


6.4 


6.6 


6.8 


7.0 


15 


5.8 


6.0 


6.2 


6.4 


6.6 


6.8 


7.1 


7.3 


7.5 


16 


6.1 


6.4 


6.6 


6.8 


7.1 


7.3 


7.5 


7.8 


8.0 


17 


6.5 


6.8 


7.0 


7.2 


7.5 


7.7 


8.0 


8.2 


8.5 


18 


6.9 


7.2 


7.4 


7.7 


7.9 


8.2 


8.5 


8.7 


9.0 


19 


7.3 


7.6 


7.8 


8.1 


8.4 


8.7 


8.9 


9.2 


9.5 


20 


7.7 


8.0 


8.2 


8.5 


8.8 


9.1 


9.4 


9.7 


10.0 


21 


8.1 


8.4 


8.7 


9.0 


9.3 


9.6 


9.9 


10.2 


10.5 


22 


8.5 


8.8 


9.1 


9.4 


9.7 


10.0 


10.3 


10.7 


11.0 


23 


8.8 


9.2 


9.5 


9.8 


10.1 


10.5 


10.8 


11.2 


11.5 


24 


9.2 


9.6 


9.9 


10.2' 


10.6 


10.9 


11.3 


11.6 


12.0 


25 


9.6 


10.0 


10.3 


10.7 


11.0 


11.4 


11.8 


12.1 


12.5 


26 


10.0 


10.4 


10.7 


11.1 


11.5 


11.8 


12.2 


12.6 


13.0 


27 


10.4 


10.8 


11.1 


11.5 


11.9 


12.3 


12.7 


13.1 


13.5 


28 


10.8 


11.1 


11.5 


11.9 


12.3 


12.8 


13.2 


13.6 


14.0 


29 


11.1 


11.5 


12.0 


12.4 


12.8 


13.2 


13.6 


14.1 


14.5 


30 


11.5 


11.9 


12.4 


12.8 


13.2 


13.7 


14.1 


14.5 


15.0 


31 


11.9 


12.3 


12.8 


13.2 


13.7 


14.1 


14.6 


15.0 


15.5 


32 


12.3 


12.7 


13.2 


13.6 


14.1 


14.6 


15.0 


15.5 


16.0 


33 


12.7 


13.1 


13.6 


14.1 


14.5 


15.0 


15.5 


16.0 


16.5 


34 


13.1 


13.5 


14.0 


14.5 


15.0 


15.5 


16.0 


16.5 


17.0 


35 


13.5 


13.9 


14.4 


14.9 


15.4 


15.9 


16.5 


17.0 


17.5 


36 


13.8 


14.3 


14.8 


15.4 


15.9 


16.4 


16.9 


17.5 


18.0 


37 


14.2 


14.7 


15.3 


15.8 


16.3 


16.8 


17.4 


17.9 


18.5 


38 


14.6 


15.1 


15.7 


16.2 


16.8 


17.3 


17.9 


18.4 


19.0 


39 


15.0 


15.5 


16.1 


16.6 


17.2 


17.8 


18.3 


18.9 


19.5 


40 


15.4 


15.9 


16.5 


17.1 


17.6 


18.2 


18.8 


19.4 


20.0 


41 


15.8 


16.3 


16.9 


17.5 


18.1 


18.7 


19.3 


19.9 


20.5 


42 


16.1 


16.7 


17.3 


17.9 


18.5 


19.1 


19.7 


20.4 


21.0 


43 


16.5 


17.1 


17.7 


18.3 


19.0 


19.6 


20.2 


209 


21.5 


44 


16.9 


17.5 


18.1 


18.8 


19.4 


20.0 


20.7 


21.3 


22.0 


45 


17.3 


17.9 


18.5 


19.2 


19.8 


20.5 


21.2 


21.8 


22.5 


46 


17.7 


18.3 


19.0 


19.6 


20.3 


20.9 


21.6 


22.3 


23.0 


47 


18.1 


18.7 


19.4 


20.0 


20.7 


21.4 


22.1 


22.8 


23.5 


48 


18.4 


19.1 


19.8 


20.5 


21.2 


21.9 


22.6 


23.3 


24.0 


49 


18.8 


19.5 


20.2 


20.9 


21.6 


22.3 


23.0 


23.8 


24.5 


50 


19.2 


19.9 


20.6 


21.3 


22.0 


22.8 


23.5 


24.2 


25.0 


100 


38.4 


39.8 


41.2 


42.6 


44.1 


45.5 


47.0 


48.5 


50.0 


200 


76.9 


79.6 


82.4 


85.3 


88.2 


91.1 


94.0 


97.0 


100.0 


300 


115.3 


119.5 


123.7 


127.9 


132.2 


136.6 


141.0 


145.5 


150.0 


400 


153.7 


159.3 


164.9 


170.6 


176.3 


182.2 


188.1 


194.0 


200.0 


500 


192.2 


199.1 


206.1 


213.2 


220.4 


227.7 


235.0 


242.5 


250.0 




1.62 


1.66 


1.70 


1.74 


1.79 


1.84 


1.89 


1.94 


2.00 




FACTOR 



To CHANGE DEP. INTO LONG. DIFF., MULTIPLY TABULAR NUMBER BY 



Table 2 



177 



To CHANGE LONG. DIFF. INTO DEP. SUBTRACT TABULAR NUMBER 
FROM LONG. DIFF. 



LONG. 
DIFF. 


MIDDLE LATITUDE 


OR 

DEP. 


52 


53 


54 


55 


56 


57 


58 


59 


60 


51 


19.6 


20.3 


21.0 


21.7 


22.5 


23.2 


24.0 


24.7 


25.5 


52 


20.0 


20.7 


21.4 


22.2 


22.9 


23.7 


24.4 


25.2 


26.0 


53 


20.4 


21.1 


21.8 


22.6 


23.4 


24.1 


24.9 


25.7 


26.5 


54 


20.8 


21.5 


22.3 


23.0 


23.8 


24.6 


25.4 


26.2 


27.0 


55 


21.1 


21.9 


22.7 


23.5 


24.2 


25.0 


25.9 


26.7 


27.5 


56 


21.5 


22.3 


23.1 


23.9 


24.7 


25.5 


26.3 


27.2 


28.0 


57 


21.9 


22.7 


23.5 


24.3 


25.1 


26.0 


26.8 


27.6 


28.5 


58 


22.3 


23.1 


23.9 


24.7 


25.6 


26.4 


27.3 


28.1 


29.0 


59 


22.7 


23.5 


24.3 


25.2 


26.0 


26.9 


27.7 


28.6 


29.5 


60 


23.1 


23.9 


24.7 


25.6 


26.4 


27.3 


28.2 


29.1 


30.0 


61 


23.4 


24.3 


25.1 


26.0 


26.9 


27.8 


28.7 


29.6 


30.5 


62 


23.8 


24.7 


25.6 


26.4 


27.3 


28.2 


29.1 


30.1 


31.0 


63 


24.2 


25.1 


26.0 


26.9 


27.8 


28.7 


29.6 


30.6 


31.5 


64 


24.6 


25.5 


26.4 


27.3 


28.2 


29.1 


30.1 


31.0 


32.0 


65 


25.0 


25.9 


26.8 


27.7 


28.7 


29.6 


30.6 


31.5 


32.5 


66 


25.4 


26.3 


27.2 


28.1 


29.1 


30.1 


31.0 


32.0 


33.0 


67 


25.8 


26.7 


27.6 


28.6 


29.5 


30.5 


31.5 


32.5 


33.5 


68 


26.1 


27.1 


28.0 


29.0 


30.0 


31.0 


32.0 


33.0 


34.0 


69 


26.5 


27.5 


28.4 


29.4 


30.4 


31.4 


32.4 


33.5 


34.5 


70 


26.9 


27.9 


28.9 


29.8 


30.9 


31.9 


32.9 


33.9 


35.0 


71 


27.3 


28.3 


29.3 


30.3 


31.3 


32.3 


33.4 


34.4 


35.5 


72 


27.7 


28.7 


29.7 


30.7 


31.7 


32.8 


33.8 


34.9 


36.0 


73 


28.1 


29.1 


30.1 


31.1 


32.2 


33.2 


34.3 


35.4 


36.5 


74 


28.4 


29.5 


30.5 


31.6 


32.6 


33.7 


34. S 


35.9 


37.0 


75 


28.8 


29.9 


30.9 


32.0 


33.1 


34.2 


35.3 


36.4 


37.5 


76 


29.2 


30.3 


31.3 


32.4 


33.5 


34.6 


35.7 


36.9 


38.0 


77 


29.6 


30.7 


31.7 


32.8 


33.9 


35.1 


36.2 


37.3 


38.5 


78 


30.0 


31.1 


32.2 


33.3 


34.4 


35.5 


36.7 


37.8 


39.0 


79 


30.4 


31.5 


32.6 


33.7 


34.8 


36.0 


37.1 


38.3 


39.5 


80 


30.7 


31.9 


33.0 


34.1 


35.3 


36.4 


37.6 


38.8 


40.0 


81 


31.1 


32.3 


33.4 


34.5 


35.7 


36.9 


38.1 


39.3 


40.5 


82 


31.5 


32.7 


33.8 


35.0 


36.1 


37.3 


38.5 


39.8 


41.0 


83 


31.9 


33.0 


34.2 


35.4 


36.6 


37.8 


39.0 


40.3 


41.5 


84 


32.3 


33.4 


34.6 


35.8 


37.0 


38.3 


39.5 


40.7 


42.0 


85 


32.7 


33.8 


35.0 


36.2 


37.5 


38.7 


40.0 


41.2 


42.5 


86 


33.1 


34.2 


35.5 


36.7 


37.9 


39.2 


40.4 


41.7 


43.0 


87 


33.4 


34.6 


35.9 


37.1 


38.4 


39.6 


40.9 


42.2 


43.5 


88 


33.8 


35.0 


36.3 


37.5 


38.8 


40.1 


41.4 


42.7 


44.0 


89 


34.2 


35.4 


36.7 


38.0 


39.2 


40.5 


41.8 


43.2 


44.5 


90 


34.6 


35.8 


37.1 


38.4 


39.7- 


41.0 


42.3 


43.6 


45.0 


91 


35.0 


36.2 


37.5 


38.8 


40.1 


41.4 


42.8 


44.1 


45.5 


92 


35.4 


36.6 


37.9 


39.2 


40.6 


41.9 


43.2 


44.6 


46.0 


93 


35.7 


37.0 


38.3 


39.7 


41.0 


42.3 


43.7 


45.1 


46.5 


94 


36.1 


37.4 


38.7 


40.1 


41.4 


42.8 


44.2 


45.6 


47.0 


95 


36.5 


37.8 


39.2 


40.5 


41.9 


43.3 


44.7 


46.1 


47.5 


96 


36.9 


38.2 


39.6 


40.9 


42.3 


43.7 


45.1 


46.6 


48.0 


97 


37.3 


38.6 


40.0 


41.4 


42.8 


44.2 


45.6 


47.0 


48.5 


98 


37.7 


39.0 


40.4 


41.8 


43.2 


44.6 


46.1 


47.5 


49.0 


90 


38.0 


39.4 


40.8 


42.2 


43.6 


45.1 


46.5 


48.0 


49.5 


100 


38.4 


39,8 


41.2 


42.6 


44.1 


45.5 


47.0 


48.5 


50.0 


600 


230.6 


238.9 


247.3 


255.9 


264.5 


273.2 


282.0 


291.0 


300.0 


700 


269.2 


279.7 


288.6 


298.5 


308.6 


318.7 


329.0 


339.6 


350.0 


800 


307.5 


319.5 


329.8 


341.2 


352.6 


364.3 


376.1 


388.0 


400.0 


900 


346.0 


358.3 


371.1 


383.8 


396.8 


409.9 


423.2 


436.6 


450.0 




1.63 


1.66 


1.70 


1.74 


1.79 


1.84 


1.89 


1.94 


2.00 




FACTOR 



To CHANGE DEP. INTO LONG. DIFF. MULTIPLY TABULAR NUMBER BY 
FACTOR AT FOOT OF COLUMN AND ADD PRODUCT TO DEP. 



178 



Table 3. Number Logarithms 








1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pts. 


100 


00000 


043 


087 


130 


173 


217 


260 


303 


346 


389 




01 


432 


475 


518 


561 


604 


647 


689 


732 


775 


817 




44 


43 


42 


02 


860 


903 


945 


988 


*030 


*072 


*115 


*157 


*199 


*242 


1 


4.4 


4.3 


4.2 


03 


01284 


326 


368 


410 


452 


494 


536 


578 


620 


662 


2 


8.8 


8.6 


8.4 


04 


703 


745 


787 


828 


870 


912 


953 


995 


*036 


*078 


3 


13.2 


12.9 


12.6 


05 


02119 


160 


202 


243 


284 


325 


366 


407 


449 


490 


4 


17.6 


17.2 


16.8 


06 


531 


572 


612 


653 


694 


735 


776 


816 


857 


898 


5 


22.0 


21.5 


21.0 
























6 


26.4 


25.8 


25.2 


07 


938 


979 


*019 


*060 


*100 


*141 


*181 


*222 


*262 


*302 


7 


30.8 


30.1 


29.4 


08 


03342 


383 


423 


463 


503 


543 


583 


623 


663 


703 


8 


35.2 


34.4 


33.6 


09 


743 


782 


822 


862 


902 


941 


981 


*021 


*060 


*100 


9 


39.6 


38.7 


37.8 


110 


04139 


179 


218 


258 


297 


336 


376 


415 


454 


493 




11 


532 


571 


610 


650 


689 


727 


766 


805 


844 


883 




41 


40 


39 


12 


922 


961 


999 


*038 


*077 


*115 


*154 


*192 


*231 


*269 


1 


4.1 


4.0 


3.9 


13 


05308 


346 


385 


423 


461 


500 


538 


576 


614 


652 


2 


8.2 


8.0 


7.8 


14 


690 


729 


767 


805 


843 


881 


918 


956 


994 


*032 


3 


12.3 


12.0 


11.7 


15 


06070 


108 


145 


183 


221 


258 


296 


333 


371 


408 


4 


16.4 


16.0 


15.6 


16 


446 


483 


521 


558 


595 


633 


670 


707 


744 


781 


5 


20.5 


20.0 


19.5 
























(> 


24.6 


24.0 


23.4 


17 


819 


856 


893 


930 


967 


*004 


*041 


*078 


*115 


*151 


7 


28.7 


28.0 


27.3 


18 


07188 


225 


262 


298 


335 


372 


408 


445 


482 


518 


8 


32.8 


32.0 


31.2 


19 


555 


591 


628 


664 


700 


737 


773 


809 


846 


882 


9 


36.9 


36.0 


35.1 


120 


918 


954 


990 


*027 


*063 


*099 


*135 


*171 


*207 


*243 




21 


08279 


314 


350 


386 


422 


458 


493 


529 


565 


600 




38 


37 


36 


22 


636 


672 


707 


743 


778 


814 


849 


884 


920 


955 


1 


3.8 


3.7 


3.6 


23 


991 


*026 


*061 


*096 


*132 


*167 


*202 


*237 


*272 


*307 


2 


7.6 


7.4 


7.2 


24 


09342 


377 


412 


447 


482 


517 


552 


587 


621 


656 


3 


11.4 


11.1 


10.8 


25 
26 


691 
10037 


726 

072 


760 
106 


795 
140 


830 
175 


864 
209 


899 
243 


934 
278 


968 
312 


*003 
346 


4 
5 
6 


15.2 
19.0 
22.8 


14.8 
18.5 
22.2 


14.4 
18.0 
21.6 


27 


380 


415 


449 


483 


517 


551 


585 


619 


653 


687 


7 


26.6 


25.9 


25.2 


28 


721 


755 


789 


823 


857 


890 


924 


958 


992 


*025 


8 


30.4 


29.6 


28.8 


29 


11059 


093 


126 


160 


193 


227 


261 


294 


327 


361 


9 


34.2 


33.3 


32.4 


130 


394 


428 


461 


494 


528 


561 


594 


628 


661 


694 




31 


727 


760 


793 


826 


860 


893 


926 


959 


992 


*024 




35 


34 


33 


32 


12057 


090 


123 


156 


189 


222 


254 


287 


320 


352 


1 


3.5 


3.4 


3.3 


33 


385 


418 


450 


483 


516 


548 


581 


613 


646 


678 


2 


7.0 


6.8 


6.6 


34 
35 
36 


710 
13033 
354 


743 
066 
386 


775 
098 
418 


808 
130 
450 


840 
162 
481 


872 
194 
513 


905 
226 
545 


937 

258 
577 


969 
290 
609 


*001 
322 
640 


3 
4 
5 
6 


10.5 
14.0 
17.5 
21.0 


10.2 
13.6 
17.0 
20.4 


9.9 
13.2 
16.5 
19.8 


37 


672 


704 


735 


767 


799 


830 


862 


893 


925 


956 


7 


24.5 


23.8 


23.1 


38 


988 


*019 


*051 


*082 


*114 


145 


*176 


*208 


*239 


*270 


8 


28.0 


27.2 


26.4 


39 


14301 


333 


364 


395 


426 


457 


489 


520 


551 


582 


9 


31.5 


30.6 


29.7 


140 


613 


644 


675 


706 


737 


768 


799 


829 


860 


891 




41 


922 


953 


983 


*014 


*045 


*076 


*106 


*137 


*168 


*198 




32 


31 


30 


42 


15229 


259 


290 


320 


351 


381 


412 


442 


473 


503 


1 


3.2 


3.1 


3.0 


43 


534 


564 


594 


625 


655 


685 


715 


746 


776 


806 


2 


6.4 


6.2 


6.0 


44 
45 
46 


836 
16137 
435 


866 
167 
465 


897 
197 
495 


927 
227 
524 


957 
256 
554 


987 
286 
584 


*017 
316 
613 


*047 
346 
643 


*077 
376 
673 


*107 
406 
702 


3 
4 
5 

6 


9.6 
12.8 
16.0 
19.2 


9.3 
12.4 
15.5 
18.6 


9.0 
12.0 
15.0 
18.0 


47 


732 


761 


791 


820 


850 


879 


909 


938 


967 


997 


7 


22.4 


21.7 


21.0 


48 


17026 


056 


085 


114 


143 


173 


202 


231 


260 


289 


8 


25.6 


24.8 


24.0 


49 


319 


348 


377 


406 


435 


464 


493 


522 


551 


580 


9 


28.8 


27.9 


27.0 


150 


609 


638 


667 


696 


725 


754 


782 


811 


840 


869 









1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pta. 



Table 3. Number Logarithms 



179 








1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pts. 


150 


17609 


638 


667 


696 


725 


754 


782 


811 


840 


869 




51 


898 


926 


955 


984 


*013 


*041 


070 


099 


*127 


*156 




52 


18184 


213 


241 


270 


298 


327 


355 


384 


412 


441 




53 


469 


498 


526 


554 


583 


611 


639 


667 


696 


724 




54 


752 


780 


808 


837 


865 


893 


921 


949 


977 


*005 




55 


19033 


061 


089 


117 


145 


173 


201 


229 


257 


285 




56 


312 


340 


368 


396 


424 


451 


479 


507 


535 


562 




57 


590 


618 


645 


673 


700 


728 


756 


783 


811 


838 




58 


866 


893 


921 


948 


976 


003 


*030 


*058 


085 


*112 




59 


20140 


167 


194 


222 


249 


276 


303 


330 


358 


385 




160 


412 


439 


466 


493 


520 


548 


575 


602 


629 


656 




61 


683 


710 


737 


763 


790 


817 


844 


871 


898 


925 




29 


28 


27 


62 


952 


978 


*005 


032 


059 


*085 


*112 


*139 


*165 


*192 


1 


2.9 


2.8 


2.7 


63 


21219 


245 


272 


299 


325 


352 


378 


405 


431 


458 


2 


5.8 


5.6 


5.4 


64 

65 
66 


484 
748 
22011 


511 
775 
037 


537 
801 
063 


564 

827 
089 


590 
854 
115 


617 
880 
141 


643 
906 
167 


669 
932 
194 


696 
958 
220 


722 
985 
246 


3 

4 
5 


8.7 
11.6 
14.5 


8.4 
11.2 
14.0 


8.1 
10.8 
13.5 
























6 


17.4 


16.8 


16.2 


67 


272 


298 


324 


350 


376 


401 


427 


453 


479 


505 


7 


20.3 


19.6 


18.9 


68 


531 


557 


583 


608 


634 


660 


686 


712 


737 


763 


8 


23.2 


22.4 


21.6 


69 


789 


814 


840 


866 


891 


917 


943 


968 


994 


*019 


9 


26.1 


25.2 


24.3 


170 


23045 


070 


096 


121 


147 


172 


198 


223 


249 


274 




71 


300 


325 


350 


376 


401 


426 


452 


477 


502 


528 




.26 


25 


24 


72 


553 


578 


603 


629 


654 


679 


704 


729 


754 


779 


1 


2.6 


2.5 


2.4 


73 


805 


830 


855 


880 


905 


930 


955 


980 


*005 


*030 


2 


5.2 


5.0 


4.8 


74 


24055 


080 


105 


130 


155 


180 


204 


229 


254 


279 


3 


7.8 


7.5 


7.2 


75 


304 


329 


353 


378 


403 


428 


452 


477 


502 


527 


4 


10.4 


10.0 


9.6 


76 


551 


576 


601 


625 


650 


674 


699 


724 


748 


773 


5 


13.0 


12.5 


12.0 
























6 


15.6 


15.0 


14.4 


77 


797 


822 


846 


871 


895 


920 


944 


969 


993 


*018 


7 


18.2 


17.5 


16.8 


78 


25042 


066 


091 


115 


139 


164 


188 


212 


237 


261 


8 


20.8 


20.0 


19.2 


79 


285 


310 


334 


358 


382 


406 


431 


455 


479 


503 


9 


23.4 


22.5 


21.6 


180 


527 


551 


575 


600 


624 


648 


672 


696 


720 


744 




81 


768 


792 


816 


840 


864 


888 


912 


935 


959 


983 




23 


22 


21 


82 


26007 


031 


055 


079 


102 


126 


150 


174 


198 


221 


1 


2.3 


2.2 


2.1 


83 


245 


269 


293 


316 


340 


364 


387 


411 


435 


458 


2 




4.4 


4.2 


84 


482 


505 


529 


553 


576 


600 


623 


647 


670 


694 


3 


6^9 


6.6 


6.3 


85 


717 


741 


764 


788 


811 


834 


858 


881 


905 


928 


4 


9.2 


8.8 


8.4 


86 


951 


975 


998 


*021 


*045 


*068 


*091 


*114 


*138 


*161 


5 


11.5 


11.0 


10.5 
























6 


13.8 


13.2 


12.6 


87 


27184 


207 


231 


254 


277 


300 


323 


346 


370 


393 


7 


16.1 


15.4 


14.7 


88 


416 


439 


462 


485 


508 


531 


554 


577 


600 


623 


8 


18.4 


17.6 


16.8 


89 


646 


669 


692 


715 


738 


761 


784 


807 


830 


852 


9 


20.7 


19.8 


18.9 


190 


875 


898 


921 


944 


967 


989 


*012 


*035 


*058 


081 




91 


28103 


126 


149 


171 


194 


217 


240 


262 


285 


307 




92 


330 


353 


375 


398 


421 


443 


466 


488 


511 


533 




93 


556 


578 


601 


623 


646 


668 


691 


713 


735 


758 




94 


780 


803 


825 


847 


870 


892 


914 


937 


959 


981 




95 


29003 


026 


048 


070 


092 


115 


137 


159 


181 


203 




96 


226 


248 


270 


292 


314 


336 


358 


380 


403 


425 




97 


447 


469 


491 


513 


535 


557 


579 


601 


623 


645 




98 


667 


688 


710 


732 


754 


776 


798 


820 


842 


863 




99 


885 


907 


929 


951 


973 


994 


016 


*038 


*060 


*081 




200 


30103 


125 


146 


168 


190 


211 


233 


255 


276 


298 









1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pts. 



180 



Table 3. Number Logarithms 








1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pts. 


200 


30103 


125 


146 


168 


190 


211 


233 


255 


276 


2<)8 










01 


320 


341 


363 


384 


406 


428 


449 


471 


492 


514 










02 


535 


557 


578 


600 


621 


643 


664 


685 


707 


728 










03 


750 


771 


792 


814 


835 


856 


878 


899 


920 


942 










04 


963 


984 


*006 


027 


*048 


*069 


*091 


*112 


*133 


*154 










05 


31175 


197 


218 


239 


260 


281 


302 


323 


345 


366 










06 


387 


408 


429 


450 


471 


492 


513 


534 


555 


576 










07 


597 


618 


639 


660 


681 


702 


723 


744 


765 


785 










08 


806 


827 


848 


869 


890 


911 


931 


952 


973 


994 










09 


32015 


035 


056 


077 


098 


118 


139 


160 


181 


201 










210 


222 


243 


263 


284 


305 


325 


346 


366 


387 


408 










11 


428 


449 


469 


490 


510 


531 


552 


572 


593 


613 




22 


21 


20 


12 


634 


654 


675 


695 


715 


736 


756 


777 


797 


818 


1 


2.2 


2.1 


2.0 


13 


838 


858 


879 


899 


919 


940 


960 


980 


*001 


*021 


2 


4.4 


4.2 


4.0 
























3 


6.6 


6.3 


6.0 


14 


33041 


062 


082 


102 


122 


143 


163 


183 


203 


224 


4 


8.8 


8.4 


8.0 


15 


244 


264 


284 


304 


325 


345 


365 


385 


405 


425 


5 


11.0 


10.5 


10.0 


16 


445 


465 


486 


506 


526 


546 


566 


586 


606 


626 


6 


13.2 


12.6 


12.0 


17 

18 
19 


646 
846 
34044 


666 
866 
064 


686 

885 
084 


706 
905 
104 


726 
925 
124 


746 
945 
143 


766 
965 
163 


786 
985 
183 


806 
*005 
203 


826 
*025 
223 


7 
8 

9 


15.4 
17.6 
19.8 


14.7 
16.8 
18.9 


14.0 
16.0 
18.0 


220 


242 


262 


282 


301 


321 


341 


361 


380 


400 


420 










21 


439 


459 


479 


498 


518 


537 


557 


577 


596 


616 










22 


635 


655 


674 


694 


713 


733 


753 


772 


792 


811 










23 


830 


850 


869 


889 


908 


928 


947 


967 


986 


*005 










24 


35025 


044 


064 


083 


102 


122 


141 


160 


180 


199 










25 


218 


238 


257 


276 


295 


315 


334 


353 


372 


392 










26 


411 


430 


449 


468 


488 


507 


526 


545 


564 


583 










27 


603 


622 


641 


660 


679 


698 


717 


736 


755 


774 










28 


793 


813 


832 


851 


870 


889 


908 


927 


946 


965 










29 


984 


*003 


*021 


*040 


*059 


*078 


*097 


*116 


*135 


*154 










230 


36173 


192 


211 


229 


248 


267 


286 


305 


324 


342 










31 


361 


380 


399 


418 


436 


455 


474 


493 


511 


530 




19 


18 


17 


32 


549 


568 


586 


605 


624 


642 


661 


680 


698 


717 


1 


1.9 


1.8 


1.7 


33 


736 


754 


773 


791 


810 


829 


847 


866 


884 


903 


2 


3.8 


3.6 


3.4 
























3 


5.7 


5.4 


5.1 


34 


922 


940 


959 


977 


996 


*014 


*033 


*051 


*070 


*088 


4 


7^6 


7.2 


6.8 


35 


37107 


125 


144 


162 


181 


199 


218 


236 


254 


273 


5 


9.5 


9.0 


8^5 


36 


291 


310 


328 


346 


365 


383 


401 


420 


438 


457 


6 


11A 


10.8 


10.2 


37 


475 


493 


511 


530 


548 


566 


585 


603 


621 


639 


7 


13.3 


12.6 


11.9 


38 


658 


676 


694 


712 


731 


749 


767 


785 


803 


822 


8 


15.2 


14.4 


13.6 


39 


840 


858 


876 


894 


912 


931 


949 


967 


985 


*003 


9 


17.1 


16.2 


15.3 


240 


38021 


039 


057 


075 


093 


112 


130 


148 


166 


184 










41 


202 


220 


238 


256 


274 


292 


310 


328 


346 


364 










42 


382 


399 


417 


435 


453 


471 


489 


507 


525 


543 










43 


561 


578 


596 


614 


632 


650 


668 


686 


703 


721 










44 


739 


757 


775 


792 


810 


828 


846 


863 


881 


899 










45 


917 


934 


952 


970 


987 


*005 


*023 


*041 


*058 


*076 










46 


39094 


111 


129 


146 


164 


182 


199 


217 


235 


252 










47 


270 


287 


305 


322 


340 


358 


375 


393 


410 


428 










48 


445 


463 


480 


498 


515 


533 


550 


568 


585 


602 










49 


620 


637 


655 


672 


690 


707 


724 


742 


759 


777 










250 


794 


811 


829 


846 


863 


881 


898 


915 


933 


950 















1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pts. 



Table 3. Number Logarithms 



181 








1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pts. 


250 


39794 


811 


829 


846 


863 


881 


898 


915 


933 


950 








51 


967 


985 


002 


*019 


*037 


*054 


*071 


*088 


*106 


*123 








52 


40140 


157 


175 


192 


209 


226 


243 


261 


278 


295 








53 


312 


329 


346 


364 


381 


398 


415 


432 


449 


466 








54 


483 


500 


518 


535 


552 


569 


586 


603 


620 


637 








55 


654 


671 


688 


705 


722 


739 


756 


773 


790 


807 








56 


824 


841 


858 


875 


892 


909 


926 


943 


960 


976 








57 


993 


*010 


027 


044 


*061 


*078 


*095 


*111 


*128 


*145 








58 


41102 


179 


196 


212 


229 


246 


263 


280 


296 


313 








59 


3:30 


347 


303 


380 


397 


414 


430 


447 


464 


481 








260 


497 


514 


531 


547 


564 


581 


597 


614 


631 


647 








61 


664 


681 


697 


714 


731 


747 


764 


780 


797 


814 


18 


17 16 


62 


830 


847 


863 


880 


896 


913 


929 


946 


963 


979 


1 1.8 


1 


.7 1.6 


63 


996 


*012 


*029 


*045 


*062 


078 


*095 


*111 


*127 


*144 


2 3.6 


3.4 3.2 


64 
65 
66 


42160 
325 

488 


177 

341 
504 


193 
357 
521 


210 
374 
537 


226 
390 
553 


243 

406 
570 


259 
423 

586 


275 
439 
602 


292 
455 
619 


308 
472 
635 


3 5.4 
4 7.2 
5 9.0 
6 10.8 


5.1 4.8 
6.8 6.4 
8.5 8.0 
10.2 9.6 


67 


651 


667 


684 


700 


716 


732 


749 


765 


781 


797 


7 12.6 


11.9 11.2 


68 


813 


830 


846 


862 


878 


894 


911 


927 


943 


959 


8 14.4 


13.6 12.8 


69 


975 


991 


*008 


*024 


040 


*056 


072 


*088 


*104 


*120 


9 16.2 


15.3 14.4 


270 


43136 


152 


169 


185 


201 


217 


233 


249 


265 


281 








71 


297 


313 


329 


345 


361 


377 


393 


409 


425 


441 








72 


457 


473 


489 


505 


521 


537 


553 


569 


584 


600 








73 


616 


632 


648 


664 


680 


696 


712 


727 


743 


759 








74 


775 


791 


807 


823 


838 


854 


870 


886 


902 


917 








75 


933 


949 


965 


981 


996 


*012 


*028 


*044 


*059 


*075 








76 


44091 


107 


122 


138 


154 


170 


185 


201 


217 


232 








77 


248 


264 


279 


295 


311 


326 


342 


358 


373 


389 








78 


404 


420 


436 


451 


467 


483 


498 


514 


529 


545 








79 


5<>0 


576 


592 


607 


623 


638 


654 


(569 


685 


700 








280 


716 


731 


747 


762 


778 


793 


809 


824 


840 


855 








81 


871 


886 


902 


917 


932 


948 


963 


979 


994 


*010 


15 


14 


82 


45025 


040 


056 


071 


086 


102 


117 


133 


148 


163 


1 1 


5 


1.4 


83 


179 


194 


209 


225 


240 


255 


271 


286 


301 


317 


2 3 




2.8 


84 


332 


347 


362 


378 


393 


408 


423 


439 


454 


469 


3 4.5 


4.2 


85 


484 


500 


515 


530 


545 


561 


576 


.591 


606 


621 


4 6.0 


5.6 


86 


637 


652 


667 


682 


697 


712 


728 


743 


758 


773 


5 7.5 


7.0 
























6 9 





8.4 


87 


788 


803 


818 


834 


849 


864 


879 


894 


909 


924 


7 10.5 


9.8 


88 


939 


954 


969 


984 


*000 


*015 


*030 


*045 


*060 


*075 


8 12.0 


11.2 


89 


46090 


105 


120 


135 


150 


165 


180 


195 


210 


225 


9 13.5 


12.6 


290 


240 


255 


270 


285 


300 


315 


330 


345 


359 


374 








91 


389 


404 


419 


434 


449 


464 


479 


494 


509 


523 








92 


538 


553 


568 


583 


598 


613 


627 


642 


657 


672 








93 


687 


702 


716 


731 


746 


761 


776 


790 


805 


820 








94 


835 


850 


864 


879 


894 


909 


923 


938 


953 


967 








95 


982 


997 


*012 


*026 


*041 


056 


*070 


*085 


*100 


*114 








96 


47129 


144 


159 


173 


188 


202 


217 


232 


246 


261 








97 


276 


290 


305 


319 


334 


349 


363 


378 


392 


407 








98 


422 


436 


451 


465 


480 


494 


509 


524 


538 


553 








99 


567 


582 


596 


611 


625 


640 


654 


669 


683 


698 








300 


712 


727 


741 


756 


770 


784 


799 


813 


828 


842 













1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pts. 



182 



Table 3. Number Logarithms 








1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pts. 


300 


47712 


727 


741 


756 


770 


784 


799 


813 


828 


842 








01 


857 


871 


885 


900 


914 


929 


943 


958 


972 


986 








02 


48001 


015 


029 


044 


058 


073 


087 


101 


116 


130 








03 


144 


159 


173 


187 


202 


216 


230 


244 


259 


273 








04 


287 


302 


316 


330 


344 


359 


373 


387 


401 


416 








05 


430 


444 


458 


473 


487 


501 


515 


530 


544 


558 








06 


572 


586 


601 


615 


629 


643 


657 


671 


686 


700 








07 


714 


728 


742 


756 


770 


785 


799 


813 


827 


841 








08 


855 


869 


883 


897 


911 


926 


940 


954 


968 


982 








09 


996 


*010 


*024 


*038 


*052 


*066 


*080 


*094 


*108 


*122 








310 


49136 


150 


164 


178 


192 


206 


220 


234 


248 


262 








11 


276 


290 


304 


318 


332 


346 


360 


374 


388 


402 




15 


14 


12 


415 


429 


443 


457 


471 


485 


499 


513 


527 


541 


1 


1.5 


1.4 


13 


554 


568 


582 


596 


610 


624 


638 


651 


665 


679 


2 


3.0 


2.8 
























3 


4.5 


4.2 


14 


693 


707 


721 


734 


748 


762 


776 


790 


803 


817 


4 


6.0 


5.6 


15 


831 


845 


859 


872 


886 


900 


914 


927 


941 


955 


5 


7.5 


7.0 


16 


969 


982 


996 


*010 


*024 


*037 


*051 


*065 


*079 


*092 


6 


9.0 


8.4 


17 


50106 


120 


133 


147 


161 


174 


188 


202 


215 


229 


7 


10.5 


9.8 


18 


243 


256 


270 


284 


297 


311 


325 


338 


352 


365 


8 


12.0 


11.2 


19 


379 


393 


406 


420 


433 


447 


461 


474 


488 


501 


9 


13.5 


12.6 


320 


515 


529 


542 


556 


569 


583 


596 


610 


623 


637 








21 


651 


664 


678 


691 


705 


718 


732 


745 


759 


772 








22 


786 


799 


813 


826 


840 


853 


866 


880 


893 


907 








23 


920 


934 


947 


961 


974 


987 


*001 


*014 


*028 


*041 








24 


51055 


068 


081 


095 


108 


121 


135 


148 


162 


175 








25 


188 


202 


215 


228 


242 


255 


268 


282 


295 


308 








26 


322 


335 


348 


362 


375 


388 


402 


415 


428 


441 








27 


455 


468 


481 


495 


508 


521 


534 


548 


561 


574 








28 


587 


601 


614 


627 


640 


654 


667 


680 


693 


706 








29 


720 


733 


746 


759 


772 


786 


799 


812 


825 


838 








330 


851 


865 


878 


891 


904 


917 


930 


943 


957 


970 








31 


983 


996 


*009 


*022 


*035 


*048 


*061 


*075 


*088 


*101 




13 


12 


32 


52114 


127 


140 


153 


166 


179 


192 


205 


218 


231 


1 


1.3 


1.2 


33 


244 


257 


270 


284 


297 


310 


323 


336 


349 


362 


2 


2.6 


2.4 
























3 


3.9 


3.6 


34 


375 


388 


401 


414 


427 


440 


453 


466 


479 


492 


4 


5.2 


4.8 


35 


504 


517 


530 


543 


556 


569 


582 


595 


608 


621 


5 


6.5 


6.0 


36 


634 


647 


660 


673 


686 


699 


711 


724 


737 


750 


6 


7.8 


7.2 


37 


763 


776 


789 


802 


815 


827 


840 


853 


866 


879 


7 


9.1 


8.4 


38 


892 


905 


917 


930 


943 


956 


969 


982 


994 


*007 


8 


10.4 


9.6 


39 


53020 


033 


046 


058 


071 


084 


097 


110 


122 


135 


9 


11.7 


10.8 


340 


148 


161 


173 


186 


199 


212 


224 


237 


250 


263 








41 


275 


288 


301 


314 


326 


339 


352 


364 


377 


390 








42 


403 


415 


428 


441 


453 


466 


479 


491 


504 


517 








43 


529 


542 


555 


567 


580 


593 


605 


618 


631 


643 








44 


656 


668 


681 


694 


706 


719 


732 


744 


757 


769 








45 


782 


794 


807 


820 


832 


845 


857 


870 


882 


895 








46 


908 


920 


933 


945 


958 


970 


983 


995 


*008 


*020 








47 


54033 


045 


058 


070 


083 


095 


108 


120 


133 


145 








48 


158 


170 


183 


195 


208 


220 


233 


245 


258 


270 








49 


283 


295 


307 


320 


332 


345 


357 


370 


382 


394 








350 


407 


419 


432 


444 


456 


469 


481 


494 


506 


518 













1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pts. 



Table 3. Number Logarithms 



183 








1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pts. 


350 


54407 


419 


432 


444 


456 


469 


481 


494 


506 


518 








51 


531 


543 


555 


568 


580 


593 


605 


617 


630 


642 








52 


654 


667 


679 


691 


704 


716 


728 


741 


753 


765 








53 


777 


790 


802 


814 


827 


839 


851 


864 


876 


888 








54 


900 


913 


925 


937 


949 


962 


974 


986 


998 


*011 








55 


55023 


035 


047 


060 


072 


084 


096 


108 


121 


133 








56 


145 


157 


169 


182 


194 


206 


218 


230 


242 


255 








57 


267 


279 


291 


303 


315 


328 


340 


352 


364 


376 








58 


388 


400 


413 


425 


437 


449 


461 


473 


485 


497 








59 


509 


522 


534 


546 


558 


570 


582 


594 


606 


618 








360 


630 


642 


654 


666 


678 


691 


703 


715 


727 


739 








61 


751 


763 


775 


787 


799 


811 


823 


835 


847 


859 




13 


12 


62 


871 


883 


895 


907 


919 


931 


943 


955 


967 


979 


1 


1.3 


1.2 


63 


991 


*003 


015 


*027 


*038 


*050 


*062 


*074 


*086 


*098 


2 


2.6 


2.4 


64 


56110 


122 


134 


146 


158 


170 


182 


194 


205 


217 


3 


3.9 


3.6 


65 


229 


241 


253 


265 


277 


289 


301 


312 


324 


336 


4 


5.2 


4.8 


66 


348 


360 


372 


384 


396 


407 


419 


431 


443 


455 


5 
6 


6.5 
7.8 


6.0 
7.2 


67 


467 


478 


490 


502 


514 


526 


538 


549 


561 


573 


7 


9.1 


8.4 


68 


585 


597 


608 


620 


632 


644 


656 


667 


679 


691 


8 


10.4 


9.6 


69 


703 


714 


726 


738 


750 


761 


773 


785 


797 


808 


9 


11.7 


10.8 


370 


820 


832 


844 


855 


867 


879 


891 


902 


914 


926 








71 


937 


949 


961 


972 


984 


996 


*008 


*019 


*031 


*043 








72 


57054 


066 


078 


089 


101 


113 


124 


136 


148 


159 








73 


171 


183 


194 


206 


217 


229 


241 


252 


264 


276 








74 


287 


299 


310 


322 


334 


345 


357 


368 


380 


392 








75 


403 


415 


426 


438 


449 


461 


473 


484 


496 


507 








76 


519 


530 


542 


553 


565 


576 


588 


600 


611 


623 








77 


634 


646 


657 


669 


680 


692 


703 


715 


726 


738 








78 


749 


761 


772 


784 


795 


807 


818 


830 


841 


852 








79 


864 


875 


887 


898 


910 


921 


933 


944 


955 


967 








380 


978 


990 


001 


*013 


*024 


*035 


*047 


058 


*070 


*081 








81 


58092 


104 


115 


127 


138 


149 


161 


172 


184 


195 




11 


10 


82 


206 


218 


229 


240 


252 


263 


274 


286 


297 


309 


1 


1.1 


1 


83 


320 


331 


343 


354 


365 


377 


388 


399 


410 


422 


2 




JL*U 

2.0 


84 


433 


444 


456 


467 


478 


490 


501 


512 


524 


535 


3 


3.3 


3.0 


85 


546 


557 


569 


580 


591 


602 


614 


625 


636 


647 


4 


4.4 


4.0 


86 


659 


670 


681 


692 


704 


715 


726 


737 


749 


760 


5 


5.5 


5.0 
























6 


6.6 


6.0 


87 


771 


782 


794 


805 


816 


827 


838 


850 


861 


872 


7 


7.7 


7.0 


88 


883 


894 


906 


917 


928 


939 


950 


961 


973 


984 


8 


8.8 


8.0 


89 


995 


*006 


017 


*028 


*040 


051 


*062 


*073 


*084 


*095 


9 


9.9 


9.0 


390 


59106 


118 


129 


140 


151 


162 


173 


184 


195 


207 








91 


218 


229 


240 


251 


262 


273 


284 


295 


306 


318 








92 


329 


340 


351 


362 


373 


384 


395 


406 


417 


428 








93 


439 


450 


461 


472 


483 


494 


506 


517 


528 


539 








94 


550 


561 


572 


583 


594 


605 


616 


627 


638 


649 








95 


660 


671 


682 


693 


704 


715 


726 


737 


748 


759 








96 


770 


780 


791 


802 


813 


824 


835 


846 


857 


868 








97 


879 


890 


901 


912 


923 


934 


945 


956 


966 


977 








98 


988 


999 


010 


*021 


032 


*043 


054 


*065 


076 


*086 








99 


60097 


108 


119 


130 


141 


152 


163 


173 


184 


195 








400 


206 


217 


228 


239 


249 


260 


271 


282 


293 


304 













1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pts. 



184 



Table 3. Number Logarithms 








1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pts. 


400 


60206 


217 


228 


239 


249 


260 


271 


282 


293 


304 




01 


314 


325 


336 


347 


358 


369 


379 


390 


401 


412 




02 


423 


433 


444 


455 


466 


477 


487 


498 


509 


520 




03 


531 


541 


552 


5(53 


574 


584 


595 


606 


617 


627 




04 


638 


649 


660 


670 


681 


692 


703 


713 


724 


735 




05 


746 


756 


767 


778 


788 


799 


810 


821 


831 


842 




06 


853 


863 


874 


885 


895 


906 


917 


927 


938 


949 




07 


959 


970 


981 


991 


*002 


*013 


*023 


*034 


*045 


*055 




08 


61066 


077 


087 


098 


109 


119 


130 


140 


151 


162 




09 


172 


183 


194 


204 


215 


225 


236 


247 


257 


268 




410 


278 


289 


300 


310 


321 


331 


342 


352 


363 


374 




11 


384 


395 


405 


416 


426 


437 


448 


458 


469 


479 




12 


490 


500 


511 


521 


532 


542 


553 


563 


574 


584 




13 


595 


606 


616 


627 


637 


648 


658 


669 


679 


690 




14 


700 


711 


721 


731 


742 


752 


763 


773 


784 


794 




15 


805 


815 


826 


836 


847 


857 


868 


878 


888 


899 




16 


909 


920 


930 


941 


951 


962 


972 


982 


993 


*003 




17 


62014 


024 


034 


045 


055 


066 


076 


086 


097 


107 




18 


118 


128 


138 


149 


159 


170 


180 


190 


201 


211 




19 


221 


232 


242 


252 


263 


273 


284 


294 


304 


315 




420 


325 


335 


346 


356 


366 


377 


387 


397 


408 


418 




21 


428 


439 


449 


459 


409 


480 


490 


500 


511 


521 


11 10 9 


22 


531 


542 


552 


562 


572 


583 


593 


603 


613 


624 


1 1.1 1.0 0.9 


23 


634 


644 


655 


<>65 


675 


685 


696 


706 


716 


726 


2 2.2 2.0 1.8 


24 


737 


747 


757 


767 


778 


788 


798 


808 


818 


829 


3 3.3 3.0 2.7 


25 


839 


849 


859 


870 


880 


890 


900 


910 


921 


931 


4 4.4 4.0 3.6 


26 


941 


951 


961 


972 


982 


992 


*002 


*012 


*022 


*033 


5 5.5 5.0 4.5 
























6 6.6 6.0 5.4 


27 


63043 


053 


063 


073 


083 


094 


104 


114 


124 


134 


7 7.7 7.0 6.3 


28 


144 


155 


165 


175 


185 


195 


205 


215 


225 


236 


8 8.8 8.0 7.2 


29 


246 


256 


266 


276 


286 


296 


306 


317 


327 


337 


9 9.9 9.0 8.1 


430 


347 


357 


367 


377 


387 


397 


407 


417 


428 


438 




31 


448 


458 


468 


478 


488 


498 


508 


518 


528 


538 




32 


548 


558 


568 


579 


589 


599 


609 


619 


629 


639 




33 


649 


659 


669 


679 


689 


699 


709 


719 


729 


739 




34 


749 


759 


769 


779 


789 


799 


809 


819 


829 


839 




35 


849 


859 


869 


879 


889 


899 


909 


919 


929 


939 




36 


949 


959 


969 


979 


988 


998 


*008 


*018 


*028 


*038 




37 


64048 


058 


068 


078 


088 


098 


108 


118 


128 


137 




38 


147 


157 


167 


177 


187 


197 


207 


217 


227 


237 




39 


246 


256 


266 


276 


286 


296 


306 


316 


326 


335 




440 


345 


355 


365 


375 


385 


395 


404 


414 


424 


434 




41 


444 


454 


464 


473 


483 


493 


503 


513 


523 


532 




42 


542 


552 


562 


572 


582 


591 


601 


611 


621 


631 




43 


640 


650 


660 


670 


680 


689 


699 


709 


719 


729 




44 


738 


748 


758 


768 


777 


787 


797 


807 


816 


826 




45 


836 


846 


856 


865 


875 


885 


895 


904 


914 


924 




46 


933 


943 


953 


963 


972 


982 


992 


*002 


*011 


*021 




47 


65031 


040 


050 


060 


070 


079 


089 


099 


108 


118 




48 


128 


137 


147 


157 


167 


176 


186 


196 


205 


215 




49 


225 


234 


244 


254 


263 


273 


283 


292 


302 


312 




450 


321 


331 


341 


350 


360 


369 


379 


389 


398 


408 









1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pts. 



Table 3. Number Logarithms 



185 








1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pts. 


450 


65 321 


331 


341 


350 


360 


369 


379 


389 


398 


408 




51 


418 


427 


437 


447 


456 


466 


475 


485 


495 


504 




52 


514 


523 


533 


543 


552 


562 


571 


581 


591 


600 




53 


610 


619 


629 


639 


648 


658 


667 


677 


686 


696 




54 


706 


715 


725 


734 


744 


753 


763 


772 


782 


792 




55 


801 


811 


820 


830 


839 


849 


858 


868 


877 


887 




56 


896 


900 


916 


925 


935 


944 


954 


963 


973 


982 




57 


992 


*001 


*011 


*020 


*030 


*039 


*049 


*058 


*068 


*077 




58 


66087 


096 


106 


115 


124 


134 


143 


153 


162 


172 




59 


181 


191 


200 


210 


219 


229 


238 


247 


257 


266 




460 


276 


285 


295 


304 


314 


323 


332 


342 


351 


361 




61 


370 


380 


389 


398 


408 


417 


427 


436 


445 


455 




62 


464 


474 


483 


492 


502 


511 


521 


530 


539 


549 




63 


558 


567 


577 


586 


596 


605 


614 


624 


633 


642 




64 


652 


661 


671 


680 


689 


699 


708 


717 


727 


736 




65 


745 


755 


764 


773 


783 


792 


801 


811 


820 


829 




66 


839 


848 


857 


867 


876 


885 


894 


904 


913 


922 




67 


932 


941 


950 


960 


969 


978 


987 


997 


*006 


*015 




68 


67 025 


034 


043 


052 


062 


071 


080 


089 


099 


108 




69 


117 


127 


136 


145 


154 


164 


173 


182 


191 


201 




470 


210 


219 


228 


237 


247 


256 


265 


274 


284 


293 




71 


302 


311 


321 


330 


339 


348 


357 


367 


376 


385 


10 9 8 


72 


394 


403 


413 


422 


431 


440 


449 


459 


468 


477 


1 1.0 0.9 0.8 


73 


486 


495 


504 


514 


523 


532 


541 


550 


560 


569 


2 2.0 1.8 1.6 


74 

75 
76 


578 
669 
761 


587 
679 
770 


596 
688 
779 


605 

697 
788 


614 
706 

797 


624 
715 

806 


633 
724 
815 


642 
733 

825 


651 
742 
834 


660 
752 
843 


3 3.0 2.7 2.4 
4 4.0 3.6 3.2 
5 5.0 4.5 4.0 
6 6.0 5.4 4.8 


77 


852 


861 


870 


879 


888 


897 


906 


916 


925 


934 


7 7.0 6.3 5.6 


78 


943 


952 


961 


970 


979 


988 


997 


*006 


*015 


*024 


8 8.0 7.2 6.4 


79 


68034 


043 


052 


061 


070 


079 


088 


097 


106 


115 


9 9.0 8.1 7.2 


480 


124 


133 


142 


151 


160 


169 


178 


187 


196 


205 




81 


215 


224 


233 


242 


251 


260 


269 


278 


287 


296 




82 


305 


314 


323 


332 


341 


350 


359 


368 


377 


386 




83 


395 


404 


413 


422 


431 


440 


449 


458 


467 


476 




84 


485 


494 


502 


511 


520 


529 


538 


547 


556 


565 




85 


574 


583 


592 


601 


610 


619 


628 


637 


646 


655 




86 


664 


673 


681 


690 


699 


708 


717 


726 


735 


744 




87 


753 


762 


771 


780 


789 


797 


806 


815 


824 


833 




88 


842 


851 


860 


869 


878 


886 


895 


904 


913 


922 




89 


931 


940 


949 


958 


9(56 


975 


984 


993 


*002 


*011 




490 


69020 


028 


037 


046 


055 


064 


073 


082 


090 


099 




91 


108 


117 


126 


135 


144 


152 


161 


170 


179 


188 




92 


197 


205 


214 


223 


232 


241 


249 


258 


267 


276 




93 


285 


294 


302 


311 


320 


329 


338 


346 


355 


364 




94 


373 


381 


390 


S99 


408 


417 


425 


434 


443 


452 




95 


461 


469 


478 


487 


496 


504 


513 


522 


531 


539 




96 


548 


557 


566 


574 


583 


592 


601 


609 


618 


627 




97 


636 


644 


653 


662 


671 


679 


688 


697 


705 


714 




98 


723 


732 


740 


749 


758 


767 


775 


784 


793 


801 




99 


810 


819 


827 


836 


845 


854 


862 


871 


880 


888 




500 


897 


906 


914 


923 


932 


940 


949 


958 


966 


975 









1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pts. 



186 



Table 3. Number Logarithms 








1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pts. 


500 


69 897 


906 


914 


923 


932 


940 


949 


958 


966 


975 




01 


984 


992 


*001 


*010 


*018 


*027 


*036 


*044 


*053 


*062 




02 


70070 


079 


088 


096 


105 


114 


122 


131 


140 


148 




03 


157 


163 


174 


183 


191 


200 


209 


217 


226 


234 




04 


243 


252 


260 


269 


278 


286 


295 


303 


312 


321 




05 


329 


338 


346 


355 


364 


372 


381 


389 


398 


406 




06 


415 


424 


432 


441 


449 


458 


467 


475 


484 


492 




07 


501 


509 


518 


526 


535 


544 


552 


561 


569 


578 




08 


586 


595 


603 


612 


621 


629 


638 


646 


655 


663 




09 


672 


680 


689 


697 


706 


714 


723 


731 


740 


749 




510 


757 


766 


774 


783 


791 


800 


808 


817 


825 


834 




11 


842 


851 


859 


8(58 


876 


885 


893 


902 


910 


919 




12 


927 


935 


944 


952 


961 


969 


978 


986 


995 


*003 




13 


71012 


020 


029 


037 


046 


054 


063 


071 


079 


088 




14 


096 


105 


113 


122 


130 


139 


147 


155 


164 


172 




15 


181 


189 


198 


206 


214 


223 


231 


240 


248 


257 




16 


265 


273 


282 


290 


299 


307 


315 


324 


332 


341 




17 


349 


357 


366 


374 


383 


391 


399 


408 


416 


425 




18 


433 


441 


450 


458 


466 


475 


483 


492 


500 


508 




19 


517 


525 


533 


542 


550 


559 


567 


575 


584 


592 




520 


600 


609 


617 


625 


'634 


642 


650 


659 


667 


675 




21 


684 


692 


700 


709 


717 


725 


734 


742 


750 


759 


987 


22 


767 


775 


784 


792 


800 


809 


817 


825 


834 


842 


1 0.9 0.8 0.7 


23 


850 


858 


867 


875 


883 


892 


900 


908 


917 


925 


2 1.8 1.6 1.4 


24 


933 


941 


950 


958 


966 


975 


983 


991 


999 


*008 


3 2.7 2.4 2.1 


25 


72016 


024 


032 


041 


049 


057 


066 


074 


082 


090 


4 3.6 3.2 2.8 


26 


099 


107 


115 


123 


132 


140 


148 


156 


165 


173 


5 4.5 4.0 3.5 
























6 5.4 4.8 4.2 


27 


181 


189 


198 


206 


214 


222 


230 


239 


247 


255 


7 6.3 5.6 4.9 


28 


263 


272 


280 


288 


296 


304 


313 


321 


329 


337 


8 7.2 6.4 5.6 


29 


346 


354 


362 


370 


378 


387 


395 


403 


411 


419 


9 8.1 7.2 6.3 


530 


428 


436 


444 


452 


460 


469 


477 


485 


493 


501 




31 


509 


518 


526 


534 


542 


550 


558 


567 


575 


583 




32 


591 


599 


607 


616 


624 


632 


640 


648 


656 


665 




33 


673 


681 


689 


697 


705 


713 


722 


730 


738 


746 




34 


754 


762 


770 


779 


787 


795 


803 


811 


819 


827 




35 


835 


843 


852 


860 


868 


876 


884 


892 


900 


908 




36 


916 


925 


933 


941 


949 


957 


965 


973 


981 


989 




37 


997 


*006 


014 


*022 


*030 


*038 


*046 


*054 


*062 


*070 




38 


73078 


086 


094 


102 


111 


119 


127 


135 


143 


151 




39 


159 


167' 


175 


183 


191 


199 


207 


215 


223 


231 




540 


239 


247 


255 


263 


272 


280 


288 


296 


304 


312 




41 


320 


328 


336 


344 


352 


360 


368 


376 


384 


392 




42 


400 


408 


416 


424 


432 


440 


448 


456 


464 


472 




43 


480 


488 


496 


504 


512 


520 


528 


536 


544 


552 




44 


560 


568 


576 


584 


592 


600 


608 


616 


624 


632 




45 


640 


648 


656 


664 


672 


679 


687 


695 


703 


711 




46 


719 


727 


735 


743 


751 


759 


767 


775 


783 


791 




47 


799 


807 


815 


823 


830 


838 


846 


854 


862 


870 




48 


878 


886 


894 


902 


910 


918 


926 


933 


941 


949 




49 


957 


965 


973 


981 


989 


997 


*005 


*013 


*020 


*028 




550 


74036 


044 


052 


060 


068 


076 


084 


092 


099 


107 









1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pts. 



Table 3. Number Logarithms 



187 








1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pts. 


550 


74036 


044 


052 


060 


068 


076 


084 


092 


099 


107 




51 


115 


123 


131 


139 


147 


155 


162 


170 


178 


186 




52 


194 


202 


210 


218 


225 


233 


241 


249 


257 


265 




53 


273 


280 


288 


296 


304 


312 


320 


327 


335 


343 




54 


351 


359 


367 


374 


382 


390 


398 


406 


414 


421 




55 


429 


437 


445 


453 


461 


468 


476 


484 


492 


500 




56 


507 


515 


523 


531 


539 


547 


554 


562 


570 


578 




57 


586 


593 


601 


609 


617 


624 


632 


640 


648 


656 




58 


663 


671 


679 


687 


695 


702 


710 


718 


726 


733 




59 


741 


749 


757 


764 


772 


780 


788 


796 


803 


811 




560 


819 


827 


834 


842 


850 


858 


865 


873 


881 


889 




61 


896 


904 


912 


920 


927 


935 


943 


950 


958 


966 




62 


974 


981 


989 


997 


*005 


*012 


*020 


028 


*035 


043 




63 


75051 


059 


066 


074 


082 


089 


097 


105 


113 


120 




64 


128 


136 


143 


151 


159 


166 


174 


182 


189 


197 




65 


205 


213 


220 


228 


236 


243 


251 


259 


266 


274 




66 


282 


289 


297 


305 


312 


320 


328 


335 


343 


351 




67 


358 


366 


374 


381 


389 


397 


404 


412 


420 


427 




68 


435 


442 


450 


458 


465 


473 


481 


488 


496 


504 




69 


511 


519 


526 


534 


542 


549 


557 


565 


572 


580 




570 


587 


595 


603 


610 


618 


626 


633 


641 


648 


656 




71 


664 


671 


679 


686 


694 


702 


709 


717 


724 


732 


8 7 


72 


740 


747 


755 


762 


770 


778 


785 


793 


800 


808 


1 0.8 0.7 


73 


815 


823 


831 


838 


846 


853 


861 


868 


876 


884 


2 1.6 1.4 


74 


891 


899 


906 


914 


921 


929 


937 


944 


952 


959 


3 2.4 2.1 


75 


967 


974 


982 


989 


997 


*005 


*012 


020 


*027 


*035 


4 3.2 2.8 


76 


76042 


050 


057 


065 


072 


080 


087 


095 


103 


110 


5 4.0 3.5 
























6 4.8 4.2 


77 


118 


125 


133 


140 


148 


155 


163 


170 


178 


185 


7 5.6 4.9 


78 


193 


200 


208 


215 


223 


230 


238 


245 


253 


260 


8 6.4 5.6 


79 


268 


275 


283 


290 


298 


305 


313 


320 


328 


335 


9 7.2 6.3 


580 


343 


350 


358 


365 


373 


380 


388 


395 


403 


410 




81 


418 


425 


433 


440 


448 


455 


462 


470 


477 


485 




82 


492 


500 


507 


515 


522 


530 


537 


545 


552 


559 




83 


567 


574 


582 


589 


597 


604 


612 


619 


626 


634 




84 


641 


649 


656 


664 


671 


678 


686 


693 


701 


708 




85 


716 


723 


730 


738 


745 


753 


760 


768 


775 


782 




86 


790 


797 


805 


812 


819 


827 


834 


842 


849 


856 




87 


864 


871 


879 


886 


893 


901 


908 


916 


923 


930 




88 


938 


945 


953 


960 


967 


975 


982 


989 


997 


004 




89 


77012 


019 


026 


034 


041 


048 


056 


063 


070 


078 




590 


085 


093 


100 


107 


115 


122 


129 


137 


144 


151 




91 


159 


166 


173 


181 


188 


195 


203 


210 


217 


225 




92 


232 


240 


247 


254 


262 


269 


276 


283 


291 


298 




93 


305 


313 


320 


327 


335 


342 


349 


357 


364 


371 




94 


379 


386 


393 


401 


408 


415 


422 


430 


437 


444 




95 


452 


459 


466 


474 


481 


488 


495 


503 


510 


517 




96 


525 


532 


539 


546 


554 


561 


568 


576 


583 


590 




97 


597 


605 


612 


619 


627 


634 


641 


648 


656 


663 




98 


670 


677 


685 


692 


699 


706 


714 


721 


728 


735 




99 


743 


750 


757 


764 


772 


779 


786 


793 


801 


808 




600 


815 


822 


830 


837 


844 


851 


859 


866 


873 


880 









1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pta. 



188 



Table 3. Number Logarithms 








1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pts. 


600 


77815 


822 


830 


837 


844 


851 


859 


8(56 


873 


880 




01 


887 


895 


902 


909 


916 


924 


931 


938 


945 


952 




02 


960 


967 


974 


981 


988 


996 


*003 


*010 


*017 


*025 




03 


78032 


039 


046 


053 


061 


068 


075 


082 


089 


097 




04 


104 


111 


118 


125 


132 


140 


147 


154 


161 


168 




05 


176 


183 


190 


197 


204 


211 


219 


226 


233 


240 




06 


247 


254 


262 


269 


276 


283 


290 


297 


305 


312 




07 


319 


326 


333 


340 


347 


355 


362 


369 


376 


383 




08 


390 


398 


405 


412 


419 


426 


433 


440 


447 


455 




09 


462 


469 


476 


483 


490 


497 


504 


512 


519 


526 




610 


533 


540 


547 


554 


561 


569 


576 


583 


590 


597 




11 


604 


611 


618 


625 


633 


640 


647 


654 


661 


668 




12 


675 


682 


689 


696 


704 


711 


718 


725 


732 


739 




13 


746 


753 


760 


767 


774 


781 


789 


796 


803 


810 




14 


817 


824 


831 


838 


845 


852 


859 


866 


873 


880 




15 


888 


895 


902 


909 


916 


923 


930 


937 


944 


951 




16 


958 


965 


972 


979 


986 


993 


*000 


*007 


*014 


*021 




17 


79029 


036 


043 


050 


057 


064 


071 


078 


085 


092 




18 


099 


106 


113 


120 


127 


134 


141 


148 


155 


162 




19 


169 


176 


183 


190 


197 


204 


211 


218 


225 


232 




620 


239 


246 


253 


260 


267 


274 


281 


288 


295 


302 




21 


309 


316 


323 


330 


337 


344 


351 


358 


365 


372 


876 


22 


379 


386 


393 


400 


407 


414 


421 


428 


435 


442 


1 0.8 0.7 0.6 


23 


449 


456 


463 


470 


477 


484 


491 


498 


505 


511 


2 1.6 1.4 1.2 


24 

25 
26 


518 

588 
657 


525 
595 
664 


532 
602 
671 


539 
609 
678 


546 
616 

685 


553 
623 
692 


560 
630 
699 


567 
637 
706 


574 
644 
713 


581 
650 
720 


3 2.4 2.1 1.8 
4 3.2 2.8 2.4 
5 4.0 3.5 3.0 
6 4.8 4.2 3.6 


27 


727 


734 


741 


748 


754 


761 


768 


775 


782 


789 


7 5.6 4.9 4.2 


28 


796 


803 


810 


817 


824 


831 


837 


844 


851 


858 


8 6.4 5.6 4.8 


29 


865 


872 


879 


886 


893 


900 


906 


913 


920 


927 


7.2 6.3 5.4 


630 


934 


941 


948 


955 


962 


969 


975 


982 


989 


996 




31 


80003 


010 


017 


024 


030 


037 


044 


051 


058 


065 




32 


072 


079 


085 


092 


099 


106 


113 


120 


127 


134 




33 


140 


147 


154 


161 


168 


175 


182 


188 


195 


202 




34 


209 


216 


223 


229 


236 


243 


250 


257 


264 


271 




35 


277 


284 


291 


298 


305 


312 


318 


325 


332 


339 




36 


346 


353 


359 


366 


373 


380 


387 


393 


400 


407 




37 


414 


421 


428 


434 


441 


448 


455 


462 


468 


475 




38 


482 


489 


496 


502 


509 


516 


523 


530 


536 


543 




39 


550 


557 


564 


570 


577 


584 


591 


598 


604 


611 




640 


618 


625 


632 


638 


645 


652 


659 


665 


672 


679 


_ 


41 


686 


693 


699 


706 


713 


720 


726 


733 


740 


747 




42 


754 


760 


767 


774 


781 


787 


794 


801 


808 


814 




43 


821 


828 


835 


841 


848 


855 


862 


868 


875 


882 




44 


889 


895 


902 


909 


916 


922 


929 


936 


943 


949 




45 


956 


963 


969 


976 


983 


990 


996 


*003 


*010 


*017 




46 


81023 


030 


037 


043 


050 


057 


064 


070 


077 


084 




47 


090 


097 


104 


111 


117 


124 


131 


137 


144 


151 




48 


158 


164 


171 


178 


184 


191 


198 


204 


211 


218 




49 


224 


231 


238 


245 


251 


258 


265 


271 


278 


285 




650 


291 


298 


305 


311 


318 


325 


331 


338 


345 


a5i 









1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pts. 



Table 3. Number Logarithms 



189 








1 


2 


3 


4 


5 | 6 


7 


8 


9 


Prop. Pts. 


650 


81291 


298 


305 


311 


318 


325 


331 


338 


345 


351 




51 


358 


365 


371 


378 


385 


391 


398 


405 


411 


418 




52 


425 


431 


438 


445 


451 


458 


465 


471 


478 


485 




53 


491 


498 


505 


511 


518 


525 


531 


538 


544 


551 




54 


558 


564 


571 


578 


584 


591 


598 


604 


611 


617 




55 


624 


631 


637 


644 


651 


657 


664 


671 


677 


684 




56 


690 


697 


704 


710 


717 


723 


730 


737 


743 


750 




57 


757 


763 


770 


776 


783 


790 


796 


803 


809 


816 




58 


823 


829 


836 


842 


849 


856 


862 


869 


875 


882 




59 


889 


895 


902 


908 


915 


921 


928 


935 


941 


948 




660 


954 


961 


968 


974 


981 


987 


994 


*000 


*007 


*014 




61 


82020 


027 


033 


040 


046 


053 


060 


066 


073 


079 




62 


086 


092 


099 


105 


112 


119 


125 


132 


138 


145 




63 


151 


158 


164 


171 


178 


184 


191 


197 


204 


210 




64 


217 


223 


230 


236 


243 


249 


256 


263 


269 


276 




65 


282 


289 


295 


302 


308 


315 


321 


328 


334 


341 




66 


347 


354 


360 


367 


373 


380 


387 


393 


400 


406 




67 


413 


419 


426 


432 


439 


445 


452 


458 


465 


471 




68 


478 


484 


491 


497 


504 


510 


517 


523 


530 


536 




69 


543 


549 


556 


562 


569 


575 


582 


5S8 


595 


601 




670 


607 


614 


620 


627 


633 


640 


646 


653 


659 


666 




71 


672 


679 


685 


692 


698 


705 


711 


718 


724 


730 


7 6 


72 


737 


743 


750 


756 


763 


769 


776 


782 


789 


795 


1 0.7 0.6 


73 


802 


808 


814 


821 


827 


834 


840 


847 


853 


860 


2 1.4 1.2 


74 


866 


872 


879 


885 


892 


898 


905 


911 


918 


924 


3 2.1 1-8 


75 


930 


937 


943 


950 


956 


963 


969 


975 


982 


988 


4 2.8 2.4 


76 


995 


*001 


*008 


*014 


*020 


*027 


*033 


*040 


*046 


*052 


5 3.5 3.0 
























6 4.2 3.6 


77 


83059 


065 


072 


078 


085 


091 


097 


104 


110 


117 


7 4.9 4.2 


78 


123 


129 


136 


142 


149 


155 


161 


168 


174 


181 


8 5.6 4.8 


79 


187 


193 


200 


206 


213 


219 


225 


232 


238 


245 


9 6.3 5.4 


680 


251 


257 


264 


270 


276 


283 


289 


296 


302 


308 




81 


315 


321 


327 


334 


340 


347 


353 


359 


366 


372 




82 


378 


385 


391 


398 


404 


410 


417 


423 


429 


436 




83 


442 


448 


455 


461 


467 


474 


480 


487 


493 


499 




84 


506 


512 


518 


525 


531 


537 


544 


550 


556 


563 




85 


569 


575 


582 


588 


594 


601 


607 


613 


620 


626 




86 


632 


639 


645 


651 


658 


664 


670 


677 


683 


689 




87 


696 


702 


708 


715 


721 


727 


734 


740 


746 


753 




88 


759 


765 


771 


778 


784 


790 


797 


803 


809 


816 




89 


822 


828 


835 


841 


847 


853 


860 


866 


872 


879 




690 


885 


891 


897 


904 


910 


916 


923 


929 


935 


942 




91 


948 


954 


960 


967 


973 


979 


985 


992 


998 


*004 




92 


84011 


017 


023 


029 


036 


042 


048 


055 


061 


067 




93 


073 


080 


086 


092 


098 


105 


111 


117 


123 


130 




94 


136 


142 


148 


155 


161 


167 


173 


180 


186 


192 




95 


198 


205 


211 


217 


223 


230 


236 


242 


248 


255 




96 


261 


267 


273 


280 


286 


292 


298 


305 


311 


317 




97 


323 


330 


336 


342 


348 


354 


361 


367 


373 


379 




98 


386 


392 


398 


404 


410 


417 


423 


429 


435 


442 




99 


448 


454 


460 


4(56 


473 


479 


485 


491 


497 


504 




700 


510 


516 


522 


528 


535 


541 


547 


553 


559 


566 









1 


2 


3 


4 


5 


6 


7 8 


9 


Prop, Pts. 



190 



Table 3. Number Logarithms 








' 1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pts. 


700 


84510 


516 


522 


528 


535 


541 


547 


553 


559 


566 




01 


572 


578 


584 


590 


597 


603 


609 


615 


621 


628 




02 


634 


640 


646 


652 


658 


665 


671 


677 


683 


689 




03 


696 


702 


708 


714 


720 


726 


733 


739 


745 


751 




04 


757 


763 


770 


776 


782 


788 


794 


800 


807 


813 




05 


819 


825 


831 


837 


844 


850 


856 


862 


868 


874 




06 


880 


887 


893 


899 


905 


911 


917 


924 


930 


936 




07 


942 


948 


954 


960 


967 


973 


979 


985 


991 


997 




08 


85003 


009 


016 


022 


028 


034 


040 


046 


052 


058 




09 


065 


071 


077 


083 


089 


095 


101 


107 


114 


120 




710 


126 


132 


138 


144 


150 


156 


163 


169 


175 


181 




11 


187 


193 


199 


205 


211 


217 


224 


230 


236 


242 




12 


248 


254 


260 


266 


272 


278 


285 


291 


297 


303 




13 


309 


315 


321 


327 


333 


339 


345 


352 


358 


364 




14 


370 


376 


382 


388 


394 


400 


406 


412 


418 


425 




15 


431 


437 


443 


449 


455 


461 


467 


473 


479 


485 




16 


491 


497 


503 


509 


516 


522 


528 


534 


540 


546 




17 


552 


558 


564 


570 


576 


582 


588 


594 


600 


606 




18 


612 


618 


625 


631 


637 


643 


649 


655 


661 


667 




19 


673 


679 


685 


691 


697 


703 


709 


715 


721 


727 




720 


733 


739 


745 


751 


757 


763 


769 


775 


781 


788 




21 


794 


800 


806 


812 


818 


824 


830 


836 


842 


848 


765 


22 


854 


860 


866 


872 


878 


884 


890 


896 


902 


908 


1 0.7 0.6 0.5 


23 


914 


920 


926 


932 


938 


944 


950 


956 


962 


968 


2 1.4 1.2 1.0 


24 
25 
26 


974 

86034 
094 


980 
040 
100 


986 
046 
106 


992 
052 
112 


998 
058 
118 


*OQ4 
064 
124 


*010 
070 
130 


*016 
076 
136 


*022 
082 
141 


*028 
088 
147 


3 2.1 1.8 1.5 
4 2.8 2.4 2.0 
5 3.5 3.0 2.5 
6 4.2 3.6 3.0 


27 


153 


159 


165 


171 


177 


183 


189 


195 


201 


207 


7 4.9 4.2 3.5 


28 


213 


219 


225 


231 


237 


243 


249 


255 


261 


267 


8 5.6 4.8 4.0 


29 


273 


279 


285 


291 


297 


303 


308 


314 


320 


326 


9 6.3 5.4 4.5 


730 


332 


338 


344 


350 


356 


362 


368 


374 


380 


386 




31 


392 


398 


404 


410 


415 


421 


427 


433 


439 


445 




32 


451 


457 


463 


469 


475 


481 


487 


493 


499 


504 




33 


510 


516 


522 


528 


534 


540 


546 


552 


558 


564 




34 


570 


576 


581 


587 


593 


599 


605 


611 


617 


623 




35 


629 


635 


641 


646 


652 


658 


664 


670 


676 


682 




36 


688 


694 


700 


705 


711 


717 


723 


729 


735 


741 




37 


747 


753 


759 


764 


770 


776 


782 


788 


794 


800 




38 


806 


812 


817 


823 


829 


835 


841 


847 


853 


859 




39 


864 


870 


876 


882 


888 


894 


900 


906 


911 


917 




740 


923 


929 


935 


941 


947 


953 


958 


964 


970 


976 




41 


982 


988 


994 


999 


*005 


*011 


*017 


*023 


*029 


*035 




42 


87040 


046 


052 


058 


064 


070 


075 


081 


087 


093 




43 


099 


105 


111 


116 


122 


128 


134 


140 


146 


151 




44 


157 


163 


169 


175 


181 


186 


192 


198 


204 


210 




45 


216 


221 


227 


233 


239 


245 


251 


256 


262 


268 




46 


274 


280 


286 


291 


297 


303 


309 


315 


320 


326 




47 


332 


338 


344 


349 


355 


361 


367 


373 


379 


384 




48 


390 


396 


402 


408 


413 


419 


425 


431 


437 


442 




49 


448 


454 


460 


466 


471 


477 


483 


489 


495 


500 




750 


506 


512 


518 


523 


529 


535 


541 


547 


552 


558 









1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pts. 



Table 3. Number Logarithms 



191 








1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pts. 


750 


87506 


512 


518 


523 


529 


535 


541 


547 


552 


558 




51 


564 


570 


576 


581 


587 


593 


599 


604 


610 


616 




52 


622 


628 


633 


639 


645 


651 


656 


662 


668 


674 




53 


679 


685 


691 


697 


703 


708 


714 


720 


726 


731 




54 


737 


743 


749 


754 


760 


766 


772 


777 


783 


789 




55 


795 


800 


806 


812 


818 


823 


829 


835 


841 


846 




56 


852 


858 


864 


869 


875 


881 


887 


892 


898 


904 




57 


910 


915 


921 


927 


933 


938 


944 


950 


955 


961 




58 


967 


973 


978 


984 


990 


996 


*001 


*007 


*013 


*018 




59 


88024 


030 


036 


041 


047 


053 


058 


064 


070 


076 




760 


081 


087 


093 


098 


104 


110 


116 


121 


127 


133 




61 


138 


144 


150 


156 


161 


167 


173 


178 


184 


190 




62 


195 


201 


207 


213 


218 


224 


230 


235 


241 


247 




63 


252 


258 


264 


270 


275 


281 


287 


292 


298 


304 




64 


309 


315 


321 


326 


332 


338 


343 


349 


355 


360 




65 


366 


372 


377 


383 


389 


395 


400 


406 


412 


417 




66 


423 


429 


434 


440 


446 


451 


457 


463 


468 


474 




67 


480 


485 


491 


497 


502 


508 


513 


519 


525 


530 




68 


536 


542 


547 


553 


559 


564 


570 


576 


581 


587 




69 


593 


598 


604 


610 


615 


621 


627 


632 


638 


643 




770 


649 


655 


660 


666 


672 


677 


683 


689 


694 


700 




71 


705 


711 


717 


722 


728 


734 


739 


745 


750 


756 


6 5 


72 


762 


767 


773 


779 


784 


790 


795 


801 


807 


812 


1 0.6 0.5 


73 


818 


824 


829 


835 


840 


846 


852 


857 


863 


868 


2 1.2 1.0 


74 
75 
76 


874 
930 
986 


880 
936 
992 


885 
941 
997 


891 
947 
*003 


897 
953 
*009 


902 
958 
*014 


908 
964 
*020 


913 
969 
*025 


919 
975 
*031 


925 
981 
*037 


3 1.8 1.5 
4 2.4 2.0 
5 3.0 2.5 
6 3.6 3.0 


77 


89042 


048 


053 


059 


064 


070 


076 


081 


087 


092 


7 4.2 3.5 


78 


098 


104 


109 


115 


120 


126 


131 


137 


143 


148 


8 4.8 4.0 


79 


154 


159 


165 


170 


176 


182 


187 


193 


198 


204 


9 5.4 4.5 


780 


209 


215 


221 


226 


232 


237 


243 


248 


254 


260 




81 


265 


271 


276 


282 


287 


293 


298 


304 


310 


315 




82 


321 


326 


332 


337 


343 


348 


354 


360 


365 


371 




83 


376 


382 


387 


393 


398 


404 


409 


415 


421 


426 




84 


432 


437 


443 


448 


454 


459 


465 


470 


476 


481 




85 


487 


492 


498 


504 


509 


515 


520 


526 


531 


537 




86 


542 


548 


553 


559 


564 


570 


575 


581 


586 


592 


1 9 


87 


597 


603 


609 


614 


620 


625 


631 


636 


642 


647 




88 


653 


658 


664 


669 


675 


680 


686 


691 


697 


702 




89 


708 


713 


719 


724 


730 


735 


741 


'746 


752 


757 




790 


763 


768 


774 


779 


785 


790 


796 


801 


807 


812 




91 


818 


823 


829 


834 


840 


845 


851 


856 


862 


867 




92 


873 


878 


883 


889 


894 


900 


905 


911 


916 


922 




93 


927 


933 


938 


944 


949 


955 


960 


966 


971 


977 




94 


982 


988 


993 


998 


*004 


*009 


*015 


*020 


*026 


*031 




95 


90037 


042 


048 


053 


059 


064 


069 


075 


080 


086 




96 


091 


097 


102 


108 


113 


119 


124 


129 


135 


140 




97 


146 


151 


157 


162 


168 


173 


179 


184 


189 


195 




98 


200 


206 


211 


217 


222 


227 


233 


238 


244 


249 




99 


255 


260 


266 


271 


276 


282 


287 


293 


298 


304 




800 


309 


314 


320 


325 


331 


336 


342 


347 


352 


358 









1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pts. 



192 



Table 3. Number Logarithms 








1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pts. 


800 


90309 


314 


320 


325 


331 


336 


342 


347 


352 


358 




01 


363 


369 


374 


380 


385 


390 


396 


401 


407 


412 




02 


417 


423 


428 


434 


439 


445 


450 


455 


461 


466 




03 


472 


477 


482 


488 


493 


499 


504 


509 


515 


520 




04 


526 


531 


536 


542 


547 


553 


558 


563 


569 


574 




05 


580 


585 


590 


596 


601 


607 


612 


617 


623 


628 




06 


634 


639 


644 


650 


655 


660 


666 


671 


677 


682 




07 


687 


693 


698 


703 


709 


714 


720 


725 


730 


736 




08 


741 


747 


752 


757 


763 


768 


773 


779 


784 


789 




09 


795 


800 


806 


811 


816 


822 


827 


832 


838 


843 




810 


849 


854 


859 


865 


870 


875 


881 


886 


891 


897 




11 


902 


907 


913 


918 


924 


929 


934 


940 


945 


950 




12 


956 


961 


966 


972 


977 


982 


988 


993 


998 


*004 




13 


91009 


014 


020 


025 


030 


036 


041 


046 


052 


057 




14 


062 


068 


073 


078 


084 


089 


094 


100 


105 


110 




15 


116 


121 


126 


132 


137 


142 


148 


153 


158 


164 




16 


169 


174 


180 


185 


190 


196 


201 


206 


212 


217 




17 


222 


228 


233 


238 


243 


249 


254 


259 


265 


270 




18 


275 


281 


286 


291 


297 


302 


307 


312 


318 


323 




19 


328 


334 


339 


344 


350 


355 


360 


365 


371 


376 




820 


381 


387 


392 


397 


403 


408 


413 


418 


424 


429 




21 


434 


440 


445 


450 


455 


461 


466 


471 


477 


482 


6 5 


22 


487 


492 


498 


503 


508 


514 


519 


524 


529 


535 


1 0.6 0.5 


23 


540 


545 


551 


556 


561 


566 


572 


577 


582 


587 


2 1.2 1.0 


24 

25 


593 
645 


598 
651 


603 
656 


609 
601 


614 
666 


619 
672 


624 

677 


630 

682 


635 

687 


640 
693 


3 1.8 1.5 
4 2.4 2.0 


26 


698 


703 


709 


714 


719 


724 


730 


735 


740 


745 


5 3.0 2.5 
6 3.6 3.0 


27 


751 


756 


761 


766 


772 


777 


782 


787 


793 


798 


7 4.2 3.5 


28 


803 


808 


814 


819 


824 


829 


834 


840 


845 


850 


8 4.8 4.0 


29 


855 


861 


866 


871 


876 


882 


887 


892 


897 


903 


9 5.4 4.5 


830 


908 


913 


918 


924 


929 


934 


939 


944 


950 


<t55 




31 


960 


965 


971 


976 


981 


986 


991 


997 


*002 


*007 




32 


92012 


018 


023 


028 


033 


038 


044 


049 


054 


059 




33 


065 


070 


075 


080 


085 


091 


096 


101 


106 


111 




34 


117 


122 


127 


132 


137 


143 


148 


153 


158 


163 




35 


169 


174 


179 


184 


189 


195 


200 


205 


210 


215 




36 


221 


226 


231 


236 


241 


247 


252 


257 


262 


267 




37 


273 


278 


283 


288 


293 


298 


304 


309 


314 


319 




38 


324 


330 


335 


340 


345 


350 


355 


361 


366 


371 




39 


376 


381 


387 


392 


3!)7 


402 


407 


412 


418 


423 




840 


428 


433 


438 


443 


449 


454 


459 


464 


469 


474 





41 


480 


485 


490 


495 


500 


505 


511 


516 


521 


526 




42 


531 


536 


542 


547 


552 


557 


562 


567 


572 


578 




43 


583 


588 


593 


598 


603 


609 


614 


619 


624 


629 




44 


634 


639 


645 


650 


655 


660 


665 


670 


675 


681 




45 


686 


691 


696 


701 


706 


711 


716 


722 


727 


732 




46 


737 


742 


747 


752 


758 


763 


768 


773 


778 


783 




47 


788 


793 


799 


804 


809 


814 


819 


824 


829 


&34 




48 


840 


845 


850 


855 


860 


865 


870 


875 


881 


886 




49 


891 


896 


901 


906 


911 


916 


921 


927 


932 


937 




850 


942 


947 


952 


957 


%2 


967 


973 


978 


983 


988 









1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pts. 



Table 3. Number Logarithms 



193 








1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pts. 


850 


92942 


947 


952 


957 


962 


967 


973 


978 


983 


988 




51 


993 


998 


003 


*008 


*013 


*018 


024 


029 


*034 


*039 




52 


93044 


049 


054 


059 


064 


069 


075 


080 


085 


090 




53 


095 


100 


105 


110 


115 


120 


125 


131 


136 


141 




54 


146 


151 


156 


161 


166 


171 


176 


181 


186 


192 




55 


197 


202 


207 


212 


217 


222 


227 


232 


237 


242 




56 


247 


252 


258 


263 


268 


273 


278 


283 


288 


293 




57 


298 


303 


308 


313 


318 


323 


328 


334 


339 


344 




58 


349 


354 


359 


364 


369 


374 


379 


384 


389 


394 


. 


59 


399 


404 


409 


414 


420 


425 


430 


435 


440 


445 




860 


450 


455 


460 


465 


470 


475 


480 


485 


490 


495 




61 


500 


505 


510 


515 


520 


526 


531 


536 


541 


546 




62 


551 


556 


561 


566 


571 


576 


581 


586 


591 


596 




63 


601 


606 


611 


616 


621 


626 


631 


636 


641 


646 




64 


651 


656 


661 


666 


671 


676 


682 


687 


692 


697 




65 


702 


707 


712 


717 


722 


727 


732 


737 


742 


747 




66 


752 


757 


762 


767 


772 


777 


782 


787 


792 


797 




67 


802 


807 


812 


817 


822 


827 


832 


837 


842 


847 




68 


852 


857 


862 


867 


872 


877 


882 


887 


892 


897 




69 


902 


907 


912 


917 


922 


927 


932 


937 


942 


947 




870 


952 


957 


962 


967 


972 


977 


982 


987 


992 


997 




71 


94002 


007 


012 


017 


022 


027 


032 


037 


042 


047 


654 


72 


052 


057 


062 


067 


072 


077 


082 


086 


091 


096 


1 0.6 0.5 0.4 


73 


101 


106 


111 


116 


121 


126 


131 


136 


141 


146 


2 1.2 1.0 0.8 


74 
75 
76 


151 

201 
250 


156 
206 
255 


161 

211 
260 


166 
216 
265 


171 

221 
270 


176 

226 
275 


181 
231 

280 


186 
236 
285 


191 

240 
290 


196 
245 
295 


3 1.8 1.5 1.2 
4 2.4 2.0 1.6 
5 3.0 2.5 2.0 
6 3.6 3.0 2.4 


77 


300 


305 


310 


315 


320 


325 


330 


335 


340 


345 


7 4.2 3.5 2.8 


78 


349 


354 


359 


364 


369 


374 


379 


384 


389 


394 


8 4.8 4.0 3.2 


79 


399 


404 


409 


414 


419 


424 


429 


433 


438 


443 


9 5.4 4.5 3.6 


880 


448 


453 


458 


463 


4(58 


473 


478 


483 


488 


493 




81 


498 


503 


507 


512 


517 


522 


527 


532 


537 


542 




82 


547 


552 


557 


5(>2 


567 


571 


576 


581 


58(5 


591 




83 


596 


601 


606 


611 


616 


621 


626 


630 


635 


640 




84 


645 


650 


655 


660 


665 


670 


675 


680 


685 


689 




85 


694 


699 


704 


709 


714 


719 


724 


729 


734 


738 




86 


743 


748 


753 


758 


763 


768 


773 


778 


783 


787 




87 


792 


797 


802 


807 


812 


817 


822 


827 


832 


836 




88 


841 


846 


851 


856 


861 


866 


871 


876 


880 


885 




89 


890 


895 


900 


905 


910 


915 


919 


924 


929 


934 




890 


939 


944 


949 


954 


959 


963 


968 


973 


978 


983 




91 


988 


993 


998 


002 


*007 


*012 


*017 


*022 


027 


*032 




92 


95036 


041 


046 


051 


056 


061 


066 


071 


075 


080 




93 


085 


090 


095 


100 


105 


109 


114 


119 


124 


129 




94 


134 


139 


143 


148 


153 


158 


163 


168 


173 


177 




95 


182 


187 


192 


197 


202 


207 


211 


216 


221 


226 




96 


231 


236 


240 


245 


250 


255 


260 


265 


270 


274 




97 


279 


284 


289 


294 


299 


303 


308 


313 


318 


323 




98 


328 


332 


337 


342 


347 


352 


357 


361 


366 


371 




99 


376 


381 


386 


390 


395 


400 


405 


410 


415 


419 




900 


424 


429, 


434 


439 


444 


448 


453 


458 


463 


468 









1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pts. 



194 



Table 3. Number Logarithms 








1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pts. 


900 


95424 


429 


434 


439 


444 


448 


453 


458 


463 


468 




01 


472 


477 


482 


487 


492 


497 


501 


506 


511 


516 




02 


521 


525 


530 


535 


540 


545 


550 


554 


559 


564 




03 


569 


574 


578 


583 


588 


593 


598 


602 


607 


612 




04 


617 


622 


626 


631 


636 


641 


646 


650 


655 


660 




05 


665 


670 


674 


679 


684 


689 


694 


698 


703 


708 




06 


713 


718 


722 


727 


732 


737 


742 


746 


751 


756 




07 


761 


766 


770 


775 


780 


785 


789 


794 


799 


804 




08 


809 


813 


818 


823 


828 


832 


837 


842 


847 


852 




09 


85H 


861 


866 


871 


875 


880 


885 


890 


895 


899 




910 


904 


909 


914 


918 


923 


928 


933 


938 


942 


947 




11 


952 


957 


961 


966 


971 


976 


980 


985 


990 


995 




12 


999 


*004 


*009 


*014 


*019 


*023 


*028 


*033 


*038 


*042 




13 


96.047 


052 


057 


061 


066 


071 


076 


080 


085 


090 




14 


095 


099 


104 


109 


114 


118 


123 


128 


133 


137 




15 


142 


147 


152 


156 


161 


166 


171 


175 


180 


185 




16 


190 


194 


199 


204 


209 


213 


218 


223 


227 


232 




17 


237 


242 


246 


251 


256 


261 


265 


270 


275 


280 




18 


284 


289 


294 


298 


303 


308 


313 


317 


322 


327 




19 


332 


336 


341 


346 


350 


355 


360 


365 


369 


374 




920 


379 


384 


388 


393 


398 


402 


407 


412 


417 


421 




21 


426 


431 


435 


440 


445 


450 


454 


459 


464 


468 


5 4 


22 


473 


478 


483 


487 


492 


497 


501 


506 


511 


515 


1 0.5 0.4 


23 


520 


525 


530 


534 


539 


544 


548 


553 


558 


562 


2. 1.0 0.8 


24 


567 


572 


577 


581 


586 


591 


595 


600 


605 


609 


3 1.5 1.2 

49 fi 1 fi 


25 


614 


619 


624 


628 


633 


638 


642 


647 


652 


656 


~.\i i ') 
50 e OH 


26 


661 


666 


670 


675 


680 


685 


689 


694 


699 


703 


_.-> '/ 

6 3.0 2.4 


27 


708 


713 


717 


722 


727 


731 


736 


741 


745 


750 


7 3.5 2.8 


28 


755 


759 


764 


769 


774 


778 


783 


788 


792 


797 


8 4.0 3.2 


29 


802 


806 


811 


816 


820 


825 


830 


834 


839 


844 


9 4.5 3.6 


930 


848 


853 


858 


862 


867 


872 


876 


881 


886 


890 




31 


895 


900 


904 


909 


914 


918 


923 


928 


932 


937 




32 


942 


946 


951 


956 


960 


965 


970 


974 


979 


984 




33 


988 


993 


997 


*002 


*007 


*011 


*016 


*021 


025 


*030 




34 


97035 


039 


044 


049 


053 


058 


063 


067 


072 


077 




35 


081 


086 


090 


095 


100 


104 


109 


114 


118 


123 




36 


128 


132 


137 


142 


146 


151 


155 


160 


165 


169 




37 


174 


179 


183 


188 


192 


197 


202 


206 


211 


216 




38 


220 


225 


230 


234 


239 


243 


248 


253 


257 


262 




39 


267 


271 


276 


280 


285 


290 


294 


299 


304 


308 




940 


313 


317 


322 


327 


331 


336 


340 


345 


350 


354 




41 


359 


364 


368 


373 


377 


382 


387 


391 


396 


400 




42 


405 


410 


414 


419 


424 


428 


433 


437 


442 


447 




43 


451 


456 


460 


465 


470 


474 


479 


483 


488 


493 




44 


497 


502 


506 


511 


516 


520 


525 


529 


534 


539 




45 


543 


548 


552 


557 


562 


566 


571 


575 


580 


585 




46 


589 


594 


598 


603 


607 


612 


617 


621 


626 


630 




47 


635 


640 


644 


649 


653 


658 


663 


667 


672 


676 




48 


681 


685 


690 


695 


699 


704 


708 


713 


717 


722 




49 


727 


731 


736 


740 


745 


749 


754 


759 


763 


768 




950 


772 


777 


782 


786 


791 


795 


800 


804 


809 


813 









1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pts. 



Table 3. Number Logarithms 



195 








1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pts. 


950 


97772 


777 


782 


786 


791 


7ft5 


800 


804 


809 


813 




51 


818 


823 


827 


832 


836 


841 


845 


850 


855 


859 




52 


864 


868 


873 


877 


882 


886 


891 


896 


900 


905 




53 


909 


914 


918 


923 


928 


932 


937 


941 


946 


950 




54 


955 


959 


964 


968 


973 


978 


982 


987 


991 


996 




55 


98000 


005 


009 


014 


019 


023 


028 


032 


037 


041 




56 


046 


050 


055 


059 


064 


068 


073 


078 


082 


087 




57 


091 


096 


100 


105 


109 


114 


118 


123 


127 


132 




58 


137 


141 


146 


150 


155 


159 


164 


168 


173 


177 




59 


182 


186 


191 


195 


200 


204 


209 


214 


218 


223 




960 


227 


232 


236 


241 


245. 


250 


254 


259 


263 


268 




61 


272 


277 


281 


286 


290 


295 


299 


304 


308 


313 




62 


318 


322 


327 


331 


336 


340 


345 


349 


354 


358 




63 


363 


367 


372 


376 


381 


385 


390 


394 


399 


403 




64 


408 


412 


417 


421 


426 


430 


435 


439 


444 


448 




65 


453 


457 


462 


466 


471 


475 


480 


484 


489 


493 




66 


498 


502 


507 


511 


516 


520 


525 


529 


534 


538 




67 


543 


547 


552 


556 


561 


565 


570 


574 


579 


583 




68 


588 


592 


597 


601 


605 


610 


614 


619 


623 


628 




69 


632 


637 


641 


646 


650 


655 


659 


664 


668 


673 




970 


677 


682 


686 


691 


695 


700 


704 


709 


713 


717 




71 


722 


726 


731 


735 


740 


744 


749 


753 


758 


762 


5 4 


72 


767 


771 


776 


780 


784 


789 


793 


798 


802 


807 


1 0.5 0.4 


73 


811 


816 


820 


825 


829 


834 


838 


843 


847 


851 


2 1.0 0.8 


74 


856 


860 


865 


869 


874 


878 


883 


887 


892 


896 


3 1.5 1.2 


75 


900 


905 


909 


914 


918 


923 


927 


932 


936 


941 


4 2.0 1.6 


76 


945 


949 


954 


958 


963 


967 


972 


976 


981 


985 


5 2.5 2.0 
























6 3.0 2.4 


77 


989 


994 


998 


*003 


007 


*012 


016 


*021 


*025 


*029 


7 3.5 2.8 


78 


99034 


038 


043 


047 


052 


056 


061 


065 


069 


074 


8 4.0 3.2 


79 


078 


083 


087 


092 


096 


100 


105 


109 


114 


118 


9 4.5 3.6 


980 


123 


127 


131 


136 


140 


145 


149 


154 


158 


162 




81 


167 


171 


176 


180 


185 


189 


193 


198 


202 


207 




82 


211 


216 


220 


224 


229 


233 


238 


242 


247 


251 




83 


255 


260 


264 


269 


273 


277 


282 


286 


291 


295 




84 


300 


304 


308 


313 


317 


322 


326 


330 


335 


339 




85 


344 


348 


352 


357 


361 


366 


370 


374 


379 


383 




86 


388 


392 


396 


401 


405 


410 


414 


419 


423 


427 




87 


432 


436 


441 


445 


449 


454 


458 


463 


467 


471 




88 


476 


480 


484 


489 


493 


498 


502 


506 


511 


515 




89 


520 


524 


528 


533 


537 


542 


546 


550 


555 


559 




990 


564 


568 


572 


577 


581 


585 


590 


594 


599 


603 




91 


607 


612 


616 


621 


625 


629 


634 


638 


642 


647 




92 


651 


656 


660 


664 


669 


673 


677 


682 


686 


691 




93 


695 


699 


704 


708 


712 


717 


721 


726 


730 


734 




94 


739 


743 


747 


752 


756 


760 


765 


769 


774 


778 




95 


782 


787 


791 


795 


800 


804 


808 


813 


817 


822 




96 


826 


830 


835 


839 


843 


848 


852 


856 


861 


865 




97 


870 


874 


878 


883 


887 


891 


896 


900 


904 


909 




98 


913 


917 


922 


926 


930 


935 


939 


944 


948 


952 




99 


957 


961 


965 


970 


974 


978 


983 


987 


991 


996 




1000 


000(10 


004 


009 


013 


017 


022 


026 


030 


035 


039 









1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Pta. 



196 



Table 4. Trigonometric Logarithms 



(180) 



(359) 179 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 










0.00 000 








0.00 000 





60 


1 


6.46 373 


.00 000 


6.46 373 


3.53 627 


.00 000 


3.53 627 


59 


2 


6.76 476 


.00 000 


6.76 476 


3.23 524 


.00 000 


.23 524 


58 


3 


6.94 085 


.00 000 


6.94 085 


3.05 915 


.00 000 


.05 915 


57 


4 


7.06 579 


.00 000 


7.06 579 


2.93 421 


.00 000 


2.93 421 


56 


5 


7.16270 


0.00 000 


7.16270 


2.83 730 


0.00 000 


2.83 730 


55 


6 


.24 188 


.00 000 


.24 188 


.75 812 


.00 000 


.75 812 


54 


7 


.30 882 


.00 000 


.30 882 


.69 118 


.00000 


.69 118 


53 


8 


.36 682 


.00 000 


.36 682 


.63 318 


.00 000 


.63 318 


52 


9 


.41 797 


.00 000 


.41 797 


.58 203 


.00 000 


.58 203 


51 


10 


7.46 373 


0.00 000 


7.46 373 


2.53 627 


0.00 000 


2.53 627 


50 


11 


.50 512 


.00 000 


.50 512 


.49 488 


.00 000 


.49 488 


49 


12 


.54 291 


.00 000 


.54 291 


.45 709 


.00 000 


.45 709 


48 


13 


.57 767 


.00 000 


.57 767 


.42 233 


.00 000 


.42 233 


47 


14 


.60 985 


.00 000 


.60 986 


.39 014 


.00 000 


.39 015 


46 


15 


7.63 982 


0.00 000 


7.63 982 


2.36018 


0.00 000 


2.36018 


45 


16 


.66 784 


.00 000 


.66 785 


.33 215 


.00 000 


.33 216 


44 


17 


.69417 


9.99 999 


.69 418 


.30 582 


.00 001 


.30 583 


43 


18 


.71 900 


.99 999 


.71 900 


.28 100 


.00 001 


.28 100 


42 


19 


.74 248 


.99 999 


.74 248 


.25 752 


.00 001 


.25 752 


41 


20 


7.76 475 


9.99 999 


7.76 476 


2.23 524 


0.00 001 


2.23 525 


40 


21 


.78 594 


.99 999 


.78 595 


.21 405 


.00 001 


.21 406 


39 


22 


.80 615 


.99 999 


.80 615 


.19 385 


.00 001 


.19 385 


38 


23 


.82 545 


.99 999 


.82 546 


.17 454 


.00 001 


.17 455 


37 


24 


.84393 


.99 999 


.84 394 


.15 606 


.00 001 


.15 607 


36 


25 


7.86 166 


9.99 999 


7.86 167 


2.13 833 


0.00 001 


2.13 834 


35 


26 


.87 870 


.99 999 


.87 871 


.12 129 


.00 001 


.12 130 


34 


27 


.89 509 


.99 999 


.89 510 


.10 490 


.00 001 


.10491 


33 


28 


.91 088 


.99 999 


.91 089 


.08911 


.00 001 


.08 912 


32 


29 


.92 612 


.99 998 


.92 613 


.07 387 


.00 002 


.07 388 


31 


30 


7.94 084 


9.99 998 


7.94 086 


2.05 914 


0.00 002 


2.05 916 


30 


31 


.95 508 


.99 998 


.95 510 


.04 490 


.00 002 


.04 492 


29 


32 


.96 887 


.99 998 


.96 889 


.03 111 


.00 002 


.03 113 


28 


33 


.98 223 


.99 998 


.98 225 


.01 775 


.00 002 


.01 777 


27 


34 


.99 520 


.99 998 


.99 522 


.00 478 


.00 002 


.00 480 


26 


35 


8.00 779 


9.99 998 


8.00 781 


1.99219 


0.00 002 


1.99 221 


25 


36 


.02 002 


.99 998 


.02 004 


.97 996 


.00 002 


.97 998 


24 


37 


.03 192 


.99 997 


.03 194 


.96 806 


.00 003 


.96 808 


23 


38 


.04350 


.99 997 


.04353 


.95 647 


.00 003 


.95 650 


22 


39 


.05 478 


.99 997 


.05 481 


.94 519 


.00 003 


.94 522 


21 


40 


8.06 578 


9.99 997 


8.06 581 


1.93419 


0.00 003 


1.93 422 


20 


41 


.07 650 


.99 997 


.07 653 


.92 347 


.00 003 


.92 350 


19 


42 


.08 696 


.99 997 


.08 700 


.91 300 


.00 003 


.91 304 


18 


43 


.09 718 


.99 997 


.09 722 


.90 278 


.00 003 


.90 282 


17 


44 


.10717 


.99 996 


.10 720 


.89 280 


.00 004 


.89 283 


16 


45 


8.11 693 


9.99 996 


8.11 696 


1.88 304 


0.00 004 


1.88 307 


15 


46 


.12 647 


.99 996 


.12 651 


.87 349 


.00 004 


.87 353 


14 


47 


.13 581 


.99 996 


.13 585 


.86 415 


.00004 


.86 419 


13 


48 


.14495 


.99 996 


.14 500 


.85 500 


.00 004 


.85 505 


12 


49 


.15391 


.99 996 


.15 395 


.84605 


.00 004 


.84609 


11 


50 


8.16 268 


9.99 995 


8.16 273 


1.83 727 


0.00 005 


1.83 732 


10 


51 


.17 128 


.99 995 


.17 133 


.82 867 


.00 005 


.82 872 


9 


52 


.17971 


.99 995 


.17 976 


.82 024 


.00 005 


.82 029 


8 


53 


.18 798 


.99 995 


.18 804 


.81 196 


.00 005 


.81 202 


7 


54 


.19610 


.99 995 


.19616 


.80 384 


.00 005 


.80 390 


6 


55 


8.20 407 


9.99 994 


8.20413 


1.79587 


0.00 006 


1.79 593 


5 


56 


.21 189 


.99 994 


.21 195 


.78 805 


.00 006 


.78811 


4 


57 


.21 958 


.99 994 


.21 964 


.78 036 


.00 006 


.78 042 


3 


58 


.22 713 


.99 994 


.22 720 


.77 280 


.00 006 


.77 287 


2 


59 


.23 456 


.99 994 


.23 462 


.76 538 


.00 006 


.76 544 


1 


60 


8.24 186 


9.99 993 


8.24 192 


1.75 808 


0.00 007 


1.75 814 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



90 (270) 



(269) 89 



Table 4. Trigonometric Logarithms 



197 



1 (181) 



(358) 178 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







8.24 186 


9.99 993 


8.24 192 


1.75 808 


0.00 007 


1.75 814 


60 


1 


.24 903 


.99 993 


.24 910 


.75 090 


.00 007 


.75 097 


59 


2 


.25 609 


.99 993 


.25 616 


.74384 


.00007 


.74 391 


58 


3 


.26 304 


.99 993 


.26 312 


.73 688 


.00 007 


.73 696 


57 


4 


.26 988 


.99 992 


.26 996 


.73 004 


.00008 


.73 012 


56 


5 


8.27 661 


9.99 992 


8.27 669 


1.72 331 


0.00 008 


1.72 339 


55 


6 


.28 324 


.99 992 


.28 332 


.71 668 


.00008 


.71 676 


54 


7 


.28 977 


.99 992 


.28 986 


.71 014 


.00008 


.71 023 


53 


8 


.29 621 


.99 992 


.29 629 


.70 371 


.00008 


.70 379 


52 


9 


.30 255 


.99 991 


.30 263 


.69 737 


.00009 


.69 745 


51 


10 


8.30 879 


9.99 991 


8.30 888 


1.69 112 


0.00 009 


1.69 121 


50 


11 


.31 495 


.99 991 


.31 505 


.68 495 


.00009 


.68 505 


49 


12 


.32 103 


.99 990 


.32 112 


.67 888 


.00010 


.67 897 


48 


13 


.32 702 


.99 990 


.32 711 


.67 289 


.00 010 


.67 298 


47 


14 


.33 292 


.99 990 


.33 302 


.66 698 


.00010 


.66 708 


46 


15 


8.33 875 


9.99 990 


8.33 886 


1.66 114 


0.00 010 


1.66 125 


45 


16 


.34 450 


.99 989 


.34 461 


.65 539 


00.011 


65550 


44 


17 


.35 018 


.99 989 


.35 029 


.64971 


.00011 


.64982 


43 


18 


.35 578 


.99 989 


.35 590 


.64410 


.00011 


.64 422 


42 


'19 


.36 131 


.99 989 


.36 143 


.63 857 


.00011 


.63 869 


41 


20 


8.36 678 


9.99 988 


8.36 689 


1.63311 


0.00 012 


1.63 322 


40 


21 


.37 217 


.99 988 


.37 229 


.62 771 


.00 012 


.62 783 


39 


22 


.37 750 


.99 988 


.37 762 


.62 238 


.00 012 


.62 250 


38 


23 


.38 276 


.99 987 


.38 289 


.61 711 


.00 013 


.61 724 


37 


24 


.38 796 


.99 987 


.38809 


.61 191 


.00 013 


.61 204 


36 


25 


8.39 310 


9.99 987 


8.39 323 


1.60 677 


0.00 013 


1.60 690 


35 


26 


.39 818 


.99 986 


.39 832 


.60 168 


.00014 


.60 182 


34 


27 


.40 320 


.99 986 


.40 334 


.59 666 


.00 014 


.59 680 


33 


28 


.40 816 


.99 986 


.40 830 


.59 170 


.00 014 


.59 184 


32 


29 


.41 307 


.99 985 


.41 321 


.58 679 


.00015 


.58 693 


31 


30 


8.41 792 


9.99 985 


8.41 807 


1.58 193 


0.00 015 


1.58 208 


30 


31 


.42 272 


.99 985 


.42287 


.57 713 


.00 015 


.57 728 


29 


32 


.42 746 


.99984 


.42 762 


.57 238 


.00 016 


.57 254 


28 


33 


.43 216 


.99984 


.43 232 


.56 768 


.00 016 


.56 784 


27 


34 


.43 680 


.99984 


.43 696 


.56 304 


.00016 


.56 320 


26 


35 


8.44 139 


9.99 983 


8.44 156 


1.55 844 


0.00 017 


1.55861 


25 


36 


.44594 


.99983 


.44611 


.55 389 


.00017 


.55 406 


24 


37 


.45044 


.99 983 


.45 061 


.54 939 


.00 017 


.54 956 


23 


38 


.45 489 


.99 982 


.45 507 


.54493 


.00018 


.54511 


22 


39 


.45 930 


.99 982 


.45 948 


.54 052 


.00018 


.54 070 


21 


40 


8.46 366 


9.99 982 


8.46 385 


1.53 615 


0.00 018 


1.53 634 


20 


41 


.46 799 


.99 981 


.46 817 


.53 183 


.00 019 


.53 201 


19 


42 


.47 226 


.99 981 


.47 245 


.52 755 


.00019 


.52 774 


18 


43 


.47 650 


.99 981 


.47 669 


.52 331 


.00019 


.52 350 


17 


44 


.48 069 


.99 980 


.48 089 


.51911 


.00020 


.51 931 


16 


45 


8.48 485 


9.99 980 


8.48 505 


1.51 495 


0.00 020 


1.51 515 


15 


46 


.48 896 


.99 979 


.48 917 


.51 083 


.00 021 


.51 104 


14 


47 


.49 304 


.99 979 


.49325 


.50 675 


.00021 


.50 696 


13 


48 


.49 708 


.99 979 


.49 729 


.50 271 


.00 021 


.50 292 


12 


49 


.50 108 


.99 978 


.50 130 


.49 870 


.00022 


.49 892 


11 


50 


8.50 504 


9.99 978 


8.50 527 


1.49 473 


0.00 022 


1.49 496 


10 


51 


.50 897 


.99 977 


.50 920 


.49 080 


.00023 


.49 103 


9 


52 


.51 287 


.99 977 


.51 310 


.48 690 


.00023 


.48713 


8 


53 


.51 673 


.99 977 


.51 696 


.48 304 


.00 023 


.48 327 


7 


54 


.52 055 


.99 976 


.52 079 


.47 921 


.00024 


.47 945 


6 


55 


8.52 434 


9.99 976 


8.52 459 


1.47 541 


0.00 024 


1.47 566 


5 


56 


.52 810 


.99 975 


.52 835 


.47 165 


.00025 


.47 190 


4 


57 


.53183 


.99 975 


.53 208 


.46 792 


.00025 


.46 817 


3 


58 


.53 552 


.99 974 


.53 578 


.46 422 


.00026 


.46448 


2 


59 


.53 919 


.99 974 


.53 945 


.46 055 


.00026 


.46 081 


1 


60 


8.54 282 


9.99 974 


8.54 308 


1.45692 


0.00 026 


1.45 718 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



91 (271) 



(268) 88 



198 



Table 4. Trigonometric Logarithms 



2 (182) 



(357) 177 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







8.54 282 


9.99 974 


8.54 308 


1.45692 


0.00 026 


1.45718 


60 


1 


.54 642 


.99 973 


.54 669 


.45 331 


.00 027 


.45 358 


59 


2 


.54 999 


.99 973 


.55 027 


.44 973 


.00 027 


.45 001 


58 


3 


.55 354 


.99 972 


.55 382 


.44 618 


.00 028 


.44 646 


57 


4 


.55 705 


.99 972 


.55 734 


.44 266 


.00 028 


.44 295 


56 


5 


8.56 054 


9.99 971 


8.56 083 


1.43917 


0.00 029 


1.43 946 


55 


6 


.56 400 


.99 971 


.56 429 


.43 571 


.00 029 


.43 600 


54 


7 


.56 743 


.99 970 


.56 773 


.43 227 


.00030 


.43 257 


53 


8 


.57 084 


.99 970 


.57 114 


.42 886 


.00 030 


.42 916 


52 


9 


.57 421 


.99 969 


.57 452 


.42 548 


.00 031 


.42 579 


51 


10 


8.57 757 


9.99 969 


8.57 788 


1.42 212 


0.00 031 


1.42 243 


50 


11 


.58 089 


.99968 


.58 121 


.41 879 


.00 032 


.41 911 


49 


12 


.58 419 


.99 968 


.58 451 


.41 549 


.00 032 


.41 581 


48 


13 


.58 747 


.99 967 


.58 779 


.41 221 


.00 033 


.41 253 


47 


14 


.59 072 


.99 967 


.59 105 


.40 895 


.00 033 


.40 928 


46 


15 


8.59 395 


9.99 967 


8.59 428 


1.40 572 


0.00 033 


1.40 605 


45 


16 


.59 715 


.99 966 


.59 749 


.40 251 


.00 034 


.40 285 


44 


17 


.60 033 


.99 966 


.60 068 


.39 932 


.00 034 


.39 967 


43 


18 


.60 349 


.99 965 


.60384 


.39616 


.00 035 


.39 651 


42 


19 


.60 662 


.99 964 


.60 698 


.39 302 


.00 036 


.39 338 


41 


20 


8.60 973 


9.99 964 


8.61 009 


1.38991 


0.00 036 


1.39 027 


40 


21 


.61 282 


.99 963 


.61 319 


.38 681 


.00 037 


.38 718 


39 


22 


.61 589 


.99 963 


.61 626 


.38 374 


.00 037 


.38411 


38 


23 


.61 894 


.99 962 


.61 931 


.38 069 


.00 038 


.38 106 


37 


24 


.62 196 


.99 962 


.62 234 


.37 766 


.00 038 


.37 804 


36 


25 


8.62 497 


9.99 961 


8.62 535 


1.37465 


0.00 039 


1.37503 


35 


26 


.62 795 


.99 961 


.62 834 


.37 166 


.00 039 


.37 205 


34 


27 


.63 091 


.99 960 


.63 131 


.36 869 


.00 940 


.36 909 


33 


28 


.63 385 


.99 960 


.63 426 


.36 574 


.00 040 


.36 615 


32 


29 


.63 678 


.99 959 


.63 718 


.36 282 


.00 041 


.36 322 


31 


30 


8.63 968 


9.99 959 


8.64 009 


1.35 991 


0.00 041 


1.36 032 


30 


31 


.64256 


.99 958 


.64298 


.35 702 


.00 042 


.35 744 


29 


32 


.64 543 


.99 958 


.64585 


.35 415 


.00 042 


.35 457 


28 


33 


.64827 


.99 957 


.64870 


.35 130 


.00043 


.35 173 


27 


34 


.65 110 


.99 956 


.65 154 


.34846 


.00044 


.34 890 


26 


35 


8.65 391 


9.99 956 


8.65 435 


1.34 565- 


0.00 044 


1.34 609 


25 


36 


.65 670 


.99 955 


.65 715 


.34285 


.00 045 


.34 330 


24 


37 


.65 947 


.99 955 


.65 993 


.34 007 


.00045 


.34 053 


23 


38 


.66 223 


.99 954 


.66 269 


.33 731 


.00 046 


.33 777 


22 


39 


.66 497 


.99 954 


.66543 


.33 457 


.00 046 


.33 503 


21 


40 


8.66 769 


9.99 953 


8.66816 


1.33 184 


0.00 047 


1.33 231 


20 


41 


.67 039 


.99 952 


.67 087 


.32 913 


.00 048 


.32 961 


19 


42 


.67 308 


.99 952 


.67 356 


.32 644 


.00 048 


.32 692 


18 


43 


.67 575 


.99 951 


.67 624 


.32 376 


.00 049 


.32 425 


17 


44 


.67 841 


.99 951 


.67 890 


.32 110 


.00 049 


.32 159 


16 


45 


8.68 104 


9.99 950 


8.68 154 


1.31 846 


0.00 050 


1.31 896 


15 


46 


.68 367 


.99 949 


.68 417 


.31 583 


.00 051 


.31 633 


14 


47 


.68 627 


.99 949 


.68 678 


.31 322 


.00 051 


.31 373 


13 


48 


.68 886 


.99 948 


.68 938 


.31 062 


.00 052 


.31 114 


12 


49 


.69 144 


.99 948 


.69 196 


.30 804 


.00 052 


.30 856 


11 


50 


8.69 400 


9.99 947 


8.69 453 


1.30 547 


0.00 053 


1.30 600 


10 


51 


.69 654 


.99 946 


, .69 708 


.30 292 


.00 054 


.30 346 


9 


52 


.69 907 


.99 946 


.69 962 


.30 038 


.00 054 


.30 093 


8 


53 


.70 159 


.99 945 


.70 214 


.29 786 


.00 055 


.29841 


7 


54 


.70 409 


.99944 


.70 465 


.29 535 


.00 056 


.29 591 


6 


55 


8.70 658 


9.99 944 


8.70 714 


1.29286 


0.00 056 


1.29 342 


5 


56 


.70 905 


.99 943 


.70 962 


.29 038 


.00 057 


.29 095 


4 


57 


.71 151 


.99 942 


.71 208 


.28 792 


.00 058 


.28849 


3 


58 


.71 395 


.99 942 


.71 453 


.28 547 


.00 058 


.28 605 


2 


59 


.71 638 


- .99 941 


.71 697 


.28 303 


.00 059 


.28 362 


1 


60 


8.71 880 


9.99 940 


8.71 940 


1.28060 


0.00 060 


1.28 120 







Cos 


Sin 


Cot 


1 Tan 


Csc 


Sec 


' 



92 (272) 



(267) 87 



Table 4. Trigonometric Logarithms 



199 



3 (183) 



(356) 176 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







8.71 880 


9.99 940 


8.71 940 


1.28060 


0.00 060 


1.28 120 


60 


1 


.72 120 


.99 940 


.72 181 


.27 819 


.00 060 


.27 880 


59 


2 


.72 359 


.99 939 


.72 420 


.27 580 


.00 061 


.27641 


58 


3 


.72 597 


.99 938 


.72 659 


.27 341 


.00062 


.27 403 


57 


4 


.72 834 


.99 938 


.72 896 


. .27 104 


.00062 


.27 166 


56 


5 


8.73 069 


9.99 937 


8.73 132 


1.26 868 


0.00 063 


1.26 931 


55 


6 


.73 303 


.99 936 


.73 366 


.26 634 


.00 064 


.26 697 


54 


7 


.73 535 


.99 936 


.73 600 


.26400 


.00 064 


.26 465 


53 


8 


.73 767 


.99 935 


.73832 


.26 168 


.00065 


.26 233 


52 


9 


.73 997 


.99 934 


.74 063 


.25 937 


.00 066 


.26 003 


51 


10 


8.74 226 


9.99 934 


8.74 292 


1.25 708 


0.00 066 


1.25 774 


50 


11 


.74 454 


.99 933 


.74 521 


.25 479 


.00067 


.25 546 


49 


12 


.74 680 


.99 932 


.74 748 


.25 252 


.00 068 


.25 320 


48 


13 


.74 906 


.99 932 


.74 974 


.25 026 


.00068 


.25 094 


47 


14 


.75 130 


.99 931 


.75 199 


.24 801 


.00069 


.24 870 


46 


15 


8.75 353 


9.99 930 


8.75 423 


1.24 577 


0.00 070 


1.24 647 


45 


16 


.75 575 


.99 929 


.75645 


.24 355 


.00071 


.24 425 


44 


17 


.75 795 


.99 929 


.75 867 


.24 133 


.00071 


.24 205 


43 


18 


.76 015 


.99 928 


.76 087 


.23 913 


.00072 


.23 985 


42 


19 


.76 234 


.99 927 


.76 306 


.23 694 


.00073 


.23 766 


41 


20 


8.76 451 


9.99 926 


8.76 525 


1.23475 


0.00 074 


1.23 549 


40 


21 


.76 667 


.99 926 


.76 742 


.23 258 


.00074 


.23 333 


39 


22 


.76 883 


.99 925 


.76 958 


.23 042 


.00 075 


.23 117 


38 


23 


.77 097 


.99 924 


.77 173 


.22 827 


.00076 


.22 903 


37 


24 


.77 310 


.99 923 


.77 387 


.22 613 


.00077 


.22 690 


36 


25 


8.77 522 


9.99 923 


8.77 600 


1.22 400 


0.00 077 


1.22 478 


35 


26 


.77 733 


.99922 


.77811 


.22 189 


.00078 


.22 267 


34 


27 


.77 943 


.99 921 


.78 022 


.21 978 


.00079 


.22 057 


33 


28 


.78 152 


.99 920 


.78 232 


.21 768 


.00080 


.21848 


32 


29 


.78 360 


.99 920 


.78441 


.21 559 


.00080 


.21 640 


31 


30 


8.78 568 


9.99 919 


8.78 649 


1.21 351 


0.00 081 


1.21 432 


30 


31 


.78 774 


.99 918 


.78 855 


.21 145 


.00082 


.21 226 


29 


32 


.78 979 


.99 917 


.79 061 


.20 939 


.00 083 


.21 021 


28 


33 


.79 183 


.99 917 


.79 266 


.20 734 


.00083 


.20 817 


27 


34 


.79 386 


.99 916 


.79 470 


.20 530 


.00084 


.20 614 


26 


35 


8.79 588 


9.99 915 


8.79 673 


1.20327 


0.00 085 


1.20412 


25 


36 


.79 789 


.99 914 


.79 875 


.20 125 


.00086 


.20211 


24 


37 


.79 990 


.99 913 


.80 076 


.19 924 


.00 087 


.20 010 


23 


38 


.80 189 


.99 913 


.80277 


.19 723 


.00 087 


.19811 


22 


39 


.80388 


.99 912 


.80476 


.19524 


.00 088 


.19612 


21 


40 


8.80 585 


9.99911 


8.80 674 


1.19326 


0.00 089 


1.19415 


20 


41 


.80 782 


.99 910 


.80 872 


.19 128 


.00090 


.19218 


19 


42 


.80 978 


.99 909 


.81 068 


.18 932 


.00 091 


.19 022 


18 


43 


.81 173 


.99 909 


.81 264 


.18 736 


.00 091 


.18 827 


17 


44 


.81 367 


.99 908 


.81 459 


.18541 


.00092 


.18 633 


16 


45 


8.81 560 


9.99 907 


8.81 653 


1.18 347 


0.00 093 


1.18440 


15 


46 


.81 752 


.99 906 


.81 846 


.18 154 


.00 094 


.18 248 


14 


47 


.81 944 


.99 905 


.82 038 


.17 962 


.00095 


.18 056 


13 


48 


.82 134 


.99904 


.82 230 


.17770 


.00096 


.17 866 


12 


49 


.82 324 


.99904 


.82 420 


.17 580 


.00096 


.17 676 


11 


50 


8.82 513 


9.99 903 


8.82 610 


1.17 390 


0.00 097 


1.17 487 


10 


51 


.82 701 


.99 902 


.82 799 


.17 201 


.00 098 


.17 299 


9 


52 


.82888 


.99 901 


.82 987 


.17013 


.00 099 


.17 112 


8 


53 


.83 075 


.99 900 


.83175 


.16 825 


.00 100 


.16 925 


7 


54 


.83261 


.99 899 


.83361 


.16 639 


.00 101 


.16 739 


6 


55 


8.83 446 


9.99 898 


8.83 547 


1.16453 


0.00 102 


1.16 554 


5 


56 


.83630 


.99 898 


.83732 


.16 268 


.00102 


.16 370 


4 


57 


.83 813 


.99 897 


.83916 


.16084 


.00103 


.16 187 


3 


58 


.83996 


.99 896 


.84100 


.15 900 


.00104 


.16 004 


2 


59 


.84177 


.99 895 


.84282 


.15718 


.00 105 


.15 823 


1 


60 


8.84 358 


9.99 894 


8.84 464 


1.15536 


0.00 106 


1.15 642 







Cos Sin 


Cot 


Tan 


Csc 


Sec 


' 



93 (273) 



(266) 86 



200 



Table 4. Trigonometric Logarithms 



4 (184) 



(355) 175 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







8.84 358 


9.99 894 


8.84464 


1.15536 


0.00 106 


1.15642 


60 


1 


.84539 


.99 893 


.84646 


.15 354 


.00 107 


.15461 


59 


2 


.84718 


.99 892 


.84826 


.15 174 


.00 108 


.15 282 


58 ' 


3 


.84897 


.99 891 


.85006 


.14 994 


.00 109 


.15 103 


57 


4 


.85 075 


.99 891 


.85 185 


.14 815 


.00 109 


.14 925 


56 


5 


8.85 252 


9.99 890 


8.85 363 


1.14 637 


0.00 110 


1.14748 


55 


6 


.85 429 


.99 889 


.85 540 


.14 460 


.00 111 


.14 571 


54 


7 


.85 605 


.99 888 


.85717 


.14 283 


.00 112 


.14 395 


53 


8 


.85 780 


.99 887 


.85 893 


.14 107 


.00 113 


.14 220 


52 


9 


.85 955 


.99 886 


.86 069 


.13 931 


.00 114 


.14 045 


51 


10 


8.86 128 


9.99 885 


8.86 243 


1.13 757 


0.00 115 


1.13 872 


50 


11 


.86 301 


.99884 


.86 417 


.13 583 


.00 116 


.13699 


49 


12 


.86 474 


.99 883 


.86 591 


.13 409 


.00 117 


.13526 


48 


13 


.86 645 


.99 882 


.86 763 


.13 237 


.00 118 


.13 355 


47 


14 


.86 816 


.99 881 


.86 935 


.13 065 


.00 119 


.13 184 


46 


15 


8.86 987 


9.99 880 


8.87 106 


1.12 894 


0.00 120 


1.13013 


45 


16 


.87 156 


.99 879 


.87 277 


.12 723 


.00 121 


.12844 


44 


17 


.87 325 


.99 879 


.87 447 


.12 553 


.00 121 


.12 675 


43 


18 


.87 494 


.99 878 


.87 616 


.12384 


.00 122 


.12 506 


42 


19 


.87 661 


.99 877 


.87 785 


.12215 


.00 123 


.12 339 


41 


20 


8.87 829 


9.99 876 


8.87 953 


1.12 047 


0.00 124 


1.12171 


40 


21 


.87 995 


.99 875 


.88 120 


.11 880 


.00 125 


.12 005 


39 


22 


.88 161 


.99 874 


.88 287 


.11 713 


00126 


.11 839 


38 


23 


.88 326 


.99 873 


.88 453 


.11 547 


.00 127 


.11 674 


37 


24 


.88 490 


.99 872 


.88 618 


.11 382 


.00 128 


.11 510 


36 


25 


8.88 654 


9.99 871 


8.88 783 


1.11217 


0.00 129 


1.11346 


35 


26 


.88817 


.99 870 


.88 948 


.11 052 


.00 130 


.11 183 


34 


27 


.88 980 


.99 869 


.89 111 


.10 889 


.00 131 


.11 020 


33 


28 


.89 142 


.99 868 


.89 274 


.10 726 


.00 132 


.10 858 


32 


29 


.89 304 


.99 867 


.89 437 


.10 563 


.00 133 


.10 696 


31 


30 


8.89 464 


9.99 866 


8.89 598 


1.10402 


0.00 134 


1.10 536 


30 


31 


.89 625 


.99 865 


.89 760 


.10 240 


.00 135 


.10375 


29 


32 


.89784 


.99 864 


.89 920 


.10 080 


.00 136 


.10216 


28 


33 


.89 943 


.99 863 


.90 080 


.09 920 


.00 137 


.10057 


27 


34 


.90 102 


.99 862 


.90 240 


.09760 


.00 138 


.09 898 


26 


35 


8.90 260 


9.99 861 


8.90 399 


1.09 601 


0.00 139 


1.09740 


25 


36 


.90 417 


.99 860 


.90 557 


.09 443 


.00 140 


.09 583 


24 


37 


.90 574 


.99 859 


.90 715 


.09 285 


.00 141 


.09 426 


23 


38 


.90 730 


.99 858 


.90 872 


.09 128 


.00 142 


.09 270 


22 


39 


.90 885 


.99 857 


.91 029 


.08 971 


.00 143 


.09 115 


21 


40 


8.91 040 


9.99 856 


8.91 185 


1.08 815 


0.00 144 


1.08 960 


20 


41 


.91 195 


.99 855 


.91 340 


.08 660 


.00 145 


.08 805 


19 


42 


.91 349 


.99 854 


.91 495 


.08 505 


.00 146 


.08 651 


18 


43 


.91 502 


.99 853 


.91 650 


.08 350 


.00 147 


.08 498 


17 


44 


.91 655 


.99 852 


.91 803 


.08 197 


.00 148 


.08 345 


16 


45 


8.91 807 


9.99 851 


8.91 957 


1.08043 


0.00 149 


1.08 193 


15 


46 


.91 959 


.99 850 


.92 110 


.07 890 


.00 150 


.08 041 


14 


47 


.92 110 


.99848 


.92 262 


.07 738 


.00 152 


.07 890 


13 


48 


.92 261 


.99 847 


.92 414 


.07 586 


.00 153 


.07 739 


12 


49 


.92411 


.99846 


.92 565 


.07 435 


.00 154 


.07 589 


11 


50 


8.92 561 


9.99 845 


8.92 716 


1.07 284 


0.00 155 


1.07439 


10 


51 


.92 710 


.99844 


.92 866 


.07 134 


.00 156 


.07 290 


9 


52 


.92 859 


.99 843 


.93 016 


.06984 


.00 157 


.07 141 


8 


53 


.93 007 


.99 842 


.93 165 


.06 835 


.00 158 


.06 993 


7 


54 


.93 154 


.99841 


.93 313 


.06 687 


.00 159 


.06846 


6 


55 


8.93 301 


9.99 840 


8.93 462 


1.06538 


0.00 160 


1.06 699 


5 


56 


.93 448 


.99 839 


.93 609 


.06 391 


.00 161 


.06 552 


4 


57 


.93 594 


.99 838 


.93 756 


.06 244 


.00 162 


.06 406 


3 


58 


.93 740 


.99 837 


.93 903 


.06 097 


.00 163 


.06 260 


2 


59 


.93 885 


.99 836 


.94 049 


.05 951 


.00 164 


.06 115 


1 


60 


8,94 030 


9.99 834 


8.94 195 


1.05 805 


0.00 166 


1.05970 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



94 (274) 



(265) 85 



Table 4. Trigonometric Logarithms 



201 



5 (185) 



(354) 174 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







8.94 030 


9.99 834 


8.94 195 


1.05&05 


0.00 166 


1.05 970 


60 


1 


.94174 


.99 833 


.94 340 


.05 660 


.00167 


.05 826 


59 


2 


.94 317 


.99 832 


.94485 


.05 515 


.00168 


.05 683 


58 


3 


.94461 


.99831 


.94 630 


.05 370 


.00 169 


.05 539 


57 


4 


.94603 


.99830 


.94 773 


.05 227 


.00170 


.05 397 


56 


5 


8.94 746 


9.99 829 


8.94 917 


1.05 083 


0.00 171 


1.05 254 


55 


6 


.94 887 


.99 828 


.95 060 


.04940 


.00172 


.05 113 


54 


7 


.95 029 


.99 827 


.95 202 


.04798 


.00173 


.04971 


53 


8 


.95 170 


.99 825 


.95344 


.04656 


.00 175 


.04830 


52 


9 


.95 310 


.99 824 


.95 486 


.04514 


.00 176 


.04690 


51 


10 


8.95 450 


9.99 823 


8.95 627 


1.04 373 


0.00 177 


1.04 550 


50 


11 


.95 589 


.99 822 


.95 767 


.04233 


.00178 


.04411 


49 


12 


.95 728 


.99 821 


.95 908 


.04092 


.00179 


.04272 


48 


13 


.95 867 


.99 820 


.96 047 


.03 953 


.00180 


.04 133 


47 


14 


.96005 


.99 819 


.96 187 


.03 813 


.00181 


.03 995 


46 


15 


8.96 143 


9.99 817 


8.96 325 


1.03 675 


0.00 183 


1.03 857 


45 


16 


.96 280 


.99 816 


.96464 


.03 536 


.00 184 


.03 720 


44 


17 


.96 417 


.99 815 


.96 602 


.03 398 


.00 185 


.03 583 


43 


18 


.96 553 


.99 814 


.96 739 


.03 261 


.00 186 


.03 447 


42 


19 


.96 689 


.99 813 


.96 877 


.03 123 


.00187 


.03 311 


41 


20 


8.96 825 


9.99 812 


8.97 013 


1.02987 


0.00 188 


1.03 175 


40 


21 


.96 960 


.99 810 


.97 150 


.02 850 


.00190 


.03040 


39 


22 


.97 095 


.99 809 


.97285 


.02 715 


.00191 


.02 905 


38 


23 


.97 229 


.99 808 


.97 421 


.02 579 


.00 192 


.02 771 


37 


24 


.97 363 


.99 807 


.97 556 


.02444 


.00193 


.02 637 


36 


25 


8.97 496 


9.99 806 


8.97 691 


1.02309 


0.00 194 


1.02 504 


35 


26 


.97 629 


.99804 


.97 825 


.02 175 


.00196 


.02 371 


34 


27 


.97 762 


.99 803 


.97 959 


.02041 


.00197 


.02 238 


33 


28 


.97 894 


.99 802 


.98 092 


.01 908 


.00198 


.02 106 


32 


29 


.98 026 


.99 801 


.98 225 


.01 775 


.00199 


.01 974 


31 


30 


8.98 157 


9.99 800 


8.98 358 


1.01 642 


0.00 200 


1.01 843 


30 


31 


.98 288 


.99 798 


.98 490 


.01 510 


.00202 


.01 712 


29 


32 


.98 419 


.99 797 


.98 622 


.01 378 


.00 203 


.01 581 


28 


33 


.98549 


.99 796 


.98 753 


.01 247 


.00204 


.01 451 


27 


34 


.98 679 


.99 795 


.98884 


.01 116 


.00205 


.01 321 


26 


35 


8.98 808 


9.99 793 


8.99 015 


1.00985 


0.00 207 


1.01 192 


25 


36 


.98 937 


.99 792 


.99 145 


.00 855 


.00208 


.01 063 


24 


37 


.99066 


.99 791 


.99 275 


.00725 


.00209 


.00934 


23 


38 


.99 194 


.99 790 


.99 405 


.00595 


.00210 


.00 806 


22 


39 


.99 322 


.99 788 


.99 534 


.00466 


.00212 


.00678 


21 


40 


8.99 450 


9.99 787 


8.99 662 


1.00338 


0.00 213 


1.00 550 


20 


41 


.99 577 


.99 786 


.99 791 


.00209 


.00214 


.00423 


19 


42 


.99704 


.99 785 


.99 919 


.00 081 


.00 215 


.00 296 


18 


43 


.99830 


.99783 


9.00 046 


0.99 954 


.00 217 


.00 170 


17 


44 


.99 956 


.99 782 


.00 174 


.99 826 


.00218 


.00044 


16 


45 


9.00 082 


9.99 781 


9.00 301 


0.99 699 


0.00 219 


0.99 918 


15 


46 


.00207 


.99780 


.00 427 


.99 573 


.00 220 


.99 793 


14 


47 


.00332 


.99 778 


.00 553 


.99 447 


.00222 


.99 668 


13 


48 


.00456 


.99 777 


.00 679 


.99 321 


.00223 


.99544 


12 


49 


.00581 


.99 776 


.00 805 


.99 195 


.00224 


.99 419 


11 


50 


9.00704 


9.99 775 


9.00 930 


0.99 070 


0.00 225 


0.99 296 


10 


51 


.00828 


.99 773 


.01 055 


.98 945 


.00227 


.99 172 


9 


52 


.00951 


.99 772 


.01 179 


.98 821 


.00228 


.99049 


8 


53 


.01 074 


.99 771 


.01 303 


.98 697 


.00229 


.98 926 


7 


54 


.01 196 


.99 769 


.01 427 


.98 573 


.00231 


.98804 


6 


55 


9.01 318 


9.99 768 


9.01 550 


0.98 450 


0.00 232 


0.98 682 


5 


56 


.01440 


.99 767 


.01 673 


.98 327 


.00233 


.98 560 


4 


57 


.01 561 


.99765 


.01 796 


.98204 


.00 235 


.98 439 


3 


58 


.01 682 


.99764 


.01 918 


.98 082 


.00 236 


.98318 


2 


59 


.01803 


.99 763 


.02040 


.97 960 


.00237 


.98 197 


1 


60 


9.01 923 


9.99 761 


9.02 162 


0.97 838 


0.00 239 


0.98 077 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



95 (275) 



(264) 84 



202 



Table 4. Trigonometric Logarithms 



6 (186) 



(353) 173 C 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







9.01 923 


9.99 761 


9.02 162 


0.97 838 


0.00 239 


0.98 077 


60 


1 


.02 043 


.99 760 


.02 283 


.97 717 


00240 


.97 957 


59 


2 


.02 163 


.99 759 


.02 404 


.97 596 


.00 241 


.97 837 


58 


3 


.02 283 


.99 757 


.02 525 


.97 475 


.00 243 


.97717 


57 


4 


.02 402 


.99 756 


.02645 


.97 355 


.00 244 


.97 598 


56 


5 


9.02 520 


9.99 755 


9.02 766 


0.97 234 


0.00 245 


0.97 480 


55 


6 


.02 639 


.99 753 


.02 885 


.97 115 


.00 247 


.97 361 


54 


7 


.02 757 


.99 752 


.03 005 


.96 995 


.00 248 


.97 243 


53 


8 


.02 874 


.99 751 


.03 124 


.96 876 


.00 249 


.97 126 


52 


9 


.02 992 


.99 749 


.03 242 


.96 758 


.00 251 


.97 008 


51 


10 


9.03 109 


9.99 748 


9.03 361 


0.96 639 


0.00 252 


0.96 891 


50 


11 


.03 226 


.99 747 


.03 479 


.96 521 


.00 253 


.96 774 


49 


12 


.03 342 


.99 745 


.03 597 


.96 403 


.00 255 


.96 658 


48 


13 


.03 458 


.99 744 


.03 714 


.96 286 


.00 256 


.96 542 


47 


14 


.03 574 


.99 742 


.03 832 


.96 168 


.00 258 


.96 426 


46 


15 


9.03 690 


9.99 741 


9.03 948 


0.96 052 


0.00 259 


0.96 310 


45 


16 


.03 805 


.99 740 


.04 065 


.95 935 


.00 260 


.96 195 


44 


17 


.03 920 


.99 738 


.04 181 


.95 819 


.00 262 


.96 080 


43 


18 


.04034 


.99 737 


.04297 


.95 703 


.00 263 


.95 966 


42 


19 


.04 149 


.99 736 


.04413 


.95 587 


.00264 


.95 851 


41 


20 


9.04 262 


9.99 734 


9.04 528 


0.95 472 


0.00 266 


0.95 738 


40 


21 


.04376 


.99 733 


.04643 


.95 357 


.00 267 


.95 624 


39 


22 


.04 490 


.99 731 


.04758 


.95 242 


.00 269 


.95 510 


38 


23 


.04 603 


.99 730 


.04873 


.95 127 


.00 270 


.95 397 


37 


24 


.04 715 


.99 728 


.04987 


.95 013 


.00 272 


.95 285 


36 


25 


9.04 828 


9.99 727 


9.05 101 


0.94 899 


0.00 273 


0.95 172 


35 


26 


.04 940 


.99 726 


.05 214 


.94 786 


.00 274 


.95 060 


34 


27 


.05 052 


.99 724 


.05 328 


.94 672 


.00 276 


.94 948 


33 


28 


.05 164 


.99 723 


.05 441 


. 94559 


.00 277 


.94 836 


32 


29 


.05 275 


.99721 


.05 553 


.94 447 


.00 279 


.94 725 


31 


30 


9.05 386 


9.99 720 


9.05 666 


0.94 334 


0.00 280 


0.94 614 


30 


31 


.05 497 


.99 718 


.05 778 


.94 222 


.00 282 


.94 503 


29 


32 


.05 607 


.99 717 


.05 890 


.94 110 


.00 283 


.94 393 


28 


33 


.05 717 


.99 716 


.06 002 


.93 998 


.00284 


.94 283 


27 


34 


.05 827 


.99 714 


.06 113 


.93 887 


.00 286 


.94 173 


26 


35 


9.05 937 


9.99 713 


9.06 224 


0.93 776 


0.00 287 


0.94 063 


25 


36 


.06 046 


.99711 


.06 335 


.93 665 


.00 289 


.93 954 


24 


37 


.06 155 


.99 710 


.06 445 


.93 555 


.00 290 


.93 845 


23 


38 


.06264 


.99 708 


.06 556 


.93 444 


.00 292 


.93 736 


22 


39 


.06 372 


.99 707 


.06 666 


.93 334 


.00 293 


.93 628 


21 


40 


9.06 481 


9.99 705 


9.06 775 


0.93 225 


0.00 295 


0.93 519 


20 


41 


.06 589 


.99 704 


.06 885 


.93 115 


.00 296 


.93411 


19 


42 


.06 696 


.99 702 


.06 994 


.93 006 


.00 298 


.93 304 


18 


43 


.06 804 


.99 701 


.07 103 


.92 897 


.00 299 


.93 196 


17 


44 


.06911 


.99 699 


.07211 


.92 789 


.00 301 


.93 089 


16 


45 


9.07 018 


9.99 698 


9.07 320 


0.92 680 


0.00 302 


0.92 982 


15 


46 


.07 124 


.99 696 


.07 428 


.92 572 


.00304 


.92 876 


14 


47 


.07 231 


.99 695 


.07 536 


.92 464 


.00 305 


.92 769 


13 


48 


.07 337 


.99 693 


.07 643 


.92 357 


00.307 


.92 663 


12 


49 


.07 442 


.99 692 


.07 751 


.92 249 


.00 308 


.92 558 


11 


50 


9.07 548 


9.99 690 


9.07 858 


0.92 142 


0.00 310 


0.92 452 


10 


51 


.07 653 


.99 689 


..07 964 


.92 036 


.00311 


.92 347 


9 


52 


.07 758 


.99 687 


.08 071 


.91 929 


.00 313 


.92 242 


8 


53 


.07 863 


.99 686 


.08 177 


.91 823 


.00 314 


.92 137 


7 


54 


.07 968 


.99 684 


.08 283 


.91 717 


.00 316 


.92 032 


6 


55 


9.08 072 


9.99 683 


9.08 389 


0.91 611 


0.00 317 


0.91 928 


5 


56 


.08 176 


.99 681 


.08 495 


.91 505 


.00 319 


.91 824 


4 


57 


.08 280 


.99 680 


.08 600 


.91 400 


.00 320 


.91 720 


3 


68 


.08 383 


.99 678 


.08 705 


.91 295 


.00 322 


.91 617 


2 


59 


.08 486 


.99 677 


.08 810 


.91 190 


.00 323 


.91 514 


1 


60 


9.08 589 


9.99 675 


9.08 914 


0.91 086 


0.00 325 


0.91 411 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



96 (276) 



(263) 83 



Table 4. Trigonometric Logarithms 



203 



7 (187) 



(352) 172 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







9.08 589 


9.99 675 


9.08 914 


0.91 086 


0.00 325 


0.91 411 


60 


1 


.08 692 


.99 674 


.09 019 


.90 981 


.00326 


.91 308 


59 


2 


.08 795 


.99 672 


.09 123 


.90 877 


.00328 


.91 205 


58 


3 


.08 897 


.99 670 


.09 227 


.90 773 


.00 330 


.91 103 


57 


4 


.08 999 


.99 669 


.09 330 


.90 670 


.00 331 


.91 001 


56 


5 


9.09 101 


9.99 667 


9.09 434 


0.90 566 


0.00 333 


0.90 899 


55 


6 


.09 202 


.99 666 


.09 537 


.90 463 


.00334 


.90 798 


54 


7 


.09 304 


.99664 


.09640 


.90 360 


.00 336 


.90 696 


53 


8 


.09 405 


.99 663 


.09 742 


.90 258 


.00337 


.90 595 


52 


9 


.09 506 


.99 661 


.09845 


.90 155 


.00 339 


.90 494 


51 


10 


9.09 606 


9.99 659 


9.09 947 


0.90 053 


0.00 341 


0.90 394 


50 


11 


.09 707 


.99 658 


.10049 


.89 951 


.00 342 


.90 293 


49 


12 


.09 807 


.99 656 


.10 150 


.89 850 


.00344 


.90 193 


48 


13 


.09907 


.99 655 


.10 252 


.89 748 


.00345 


.90093 


47 


14 


.10 006 


.99 653 


.10353 


.89647 


.00347 


.89 994 


46 


15 


9.10 106 


9.99 651 


9.10 454 


0.89 546 


0.00 349 


0.89 894 


45 


16 


.10 205 


.99 650 


.10 555 


.89 445 


.00350 


.89 795 


44 


17 


.10 304 


.99648 


.10 656 


.89 344 


.00 352 


.89 696 


43 


18 


.10 402 


.99647 


.10 756 


.89 244 


.00 353 


.89 598 


42 


19 


.10 501 


.99645 


.10 856 


.89 144 


.00 355 


.89 499 


41 


20 


9.10599 


9.99 643 


9.10 956 


0.89044 


0.00 357 


0.89 401 


40 


21 


.10 697 


.99 642 


.11 056 


.88 944 


.00 358 


.89 303 


39 


22 


.10 795 


.99640 


.11 155 


.88845 


.00360 


.89 205 


38 


23 


.10 893 


.99 638 


.11254 


.88 746 


.00362 


.89 107 


37 


24 


.10 990 


.99 637 


.11 353 


.88647 


.00 363 


.89 010 


36 


25 


9.11 087 


9.99 635 


9.11452 


0.88 548 


0.00 365 


0.88913 


35 


26 


.11 184 


.99 633 


.11 551 


.88449 


.00 367 


.88 816 


34 


27 


.11 281 


.99 632 


.11 649 


.88351 


.00368 


.88 719 


33 


28 


.11 377 


.99 630 


.11 747 


.88253 


.00 370 


.88623 


32 


29 


.11 474 


.99 629 


.11 845 


.88155 


.00371 


.88 526 


31 


30 


9.11 570 


9.99 627 


9.11 943 


0.88 057 


0.00 373 


0.88 430 


30 


31 


.11 666 


.99 625 


.12040 


.87 960 


.00375 


.88 334 


29 


32 


.11761 


.99 624 


.12 138 


.87 862 


.00376 


.88239 


28 


33 


.11 857 


.99 622 


.12 235 


.87 765 


.00378 


.88 143 


27 


34 


.11 952 


.99 620 


.12 332 


.87 668 


.00380 


.88 048 


26 


35 


9.12 047 


9.99 618 


9.12 428 


0.87 572 


0.00 382 


0.87 953 


25 


36 


.12 142 


.99 617 


.12 525 


.87 475 


.00 383 


.87 858 


24 


37 


.12 236 


.99 615 


.12 621 


.87 379 


.00 385 


.87764 


23 


38 


.12331 


.99 613 


.12717 


.87283 


.00387 


.87 669 


22 


39 


.12 425 


.99 612 


.12 813 


.87 187 


.00388 


.87 575 


21 


40 


9.12519 


9.99 610 


9.12 909 


0.87 091 


0.00 390 


0.87 481 


20 


41 


.12612 


.99 608 


.13004 


.86 996 


.00 392 


.87 388 


19 


42 


.12 706 


.99 607 


.13 099 


.86 901 


.00 393 


.87 294 


18 


43 


.12 799 


.99 605 


.13 194 


.86806 


.00395 


.87 201 


17 


44 


.12 892 


.99 603 


.13 289 


.86 711 


.00397 


.87 108 


16 


45 


9.12 985 


9.99 601 


9.13 384 


0.86 616 


0.00 399 


0.87 015 


15 


46 


.13 078 


.99 600 


.13 478 


.86 522 


.00 400 


.86 922 


14 


47 


.13 171 


.99 598 


.13 573 


.86 427 


.00402 


.86 829 


13 


48 


.13 263 


.99 596 


.13 667 


.86 333 


.00 404 


.86 737 


12 


49 


.13355 


.99 595 


.13 761 


.86 239 


.00 405 


.86645 


11 


50 


9.13447 


9.99 593 


9.13854 


0.86 146 


0.00 407 


0.86 553 


10 


51 


.13 539 


.99 591 


.13 948 


.86 052 


.00409 


.86 461 


9 


52 


.13 630 


.99 589 


.14041 


.85959 


.00411 


.86 370 


8 


53 


.13 722 


.99 588 


.14 134 


.85866 


.00412 


.86 278 


7 


54 


.13 813 


.99 586 


.14 227 


.85 773 


.00 414 


.86 187 


6 


55 


9.13 904 


9.99 584 


9.14 320 


0.85 680 


0.00416 


0.86 096 


5 


56 


.13 994 


.99 582 


.14412 


.85588 


.00 418 


.86 006 


4 


57 


.14 085 


.99 581 


.14 504 


.85496 


.00 419 


.85 915 


3 


58 


.14 175 


.99 579 


.14 597 


.85 403 


.00 421 


.85 825 


2 


59 


.14 266 


.99 577 


.14 688 


.85312 


.00 423 


.85 734 


1 


60 


9.14356 


9.99 575 


9.14 780 


0.85 220 


0.00 425 


0.85 644 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



97 (277) 



(262) 82 



204 



Table 4. Trigonometric Logarithms 



8 (188) 



(351) 171 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







9.14356 


9.99 575 


9.14 780 


0.85 220 


0.00 425 


0.85 644 


60 


1 


.14445 


.99 574 


.14 872 


.85 128 


.00 426 


.85 555 


59 


2 


.14 535 


.99 572 


.14 963 


.85 037 


.00 428 


.85465 


58 


3 


.14 624 


.99 570 


.15 054 


.84946 


.00 430 


.85 376 


57 


4 


.14714 


.99 568 


.15 145 


.84855 


.00432 


.85 286 


56 


5 


9.14 803 


9.99 566 


9.15 236 


0.84764 


0.00 434 


0.85 197 


55 


6 


.14891 


.99 565 


.15327 


.84 673 


.00 435 


.85 109 


54 


7 


.14 980 


.99 563 


.15417 


.84583 


.00 437 


.85 020 


53 


8 


.15 069 


.99 561 


.15 508 


.84492 


.00 439 


.84931 


52 


9 


.15 157 


.99 559 


.15 598 


.84402 


.00 441 


.84843 


51 


10 


9.15 245 


9.99 557 


9.15 688 


0.84 312 


0.00 443 


0.84 755 


50 


11 


.15 333 


.99 556 


.15 777 


.84223 


.00 444 


.84667 


49 


12 


.15421 


.99 554 


.15 867 


.84 133 


.00 446 


.84579 


48 


13 


.15 508 


.99 552 


.15 956 


.84044 


.00 448 


.84 492 


47 


14 


.15 596 


.99 550 


.16 046 


.83 954 


.00 450 


.84404 


46 


15 


9.15 683 


9.99 548 


9.16 135 


0.83 865 


0.00 452 


0.84 317 


45 


16 


.15770 


.99 546 


.16 224 


.83776 


.00 454 


.84230 


44 


17 


.15 857 


.99 545 


.16312 


.83 688 


.00 455 


.84 143 


43 


18 


.15 944 


.99 543 


.16401 


.83 599 


.00457 


.84056 


42 


19 


.16 030 


.99 541 


.16489 


.83511 


.00 459 


.83 970 


41 


20 


9.16116 


9.99 539 


9.16 577 


0.83 423 


0.00 461 


0.83 884 


40 


21 


.16 203 


.99 537 


.16 665 


.83 335 


.00463 


.83 797 


39 


22 


.16 289 


.99 535 


.16753 


.83 247 


.00 465 


.83711 


38 


23 


.16374 


.99 533 


.16841 


.83 159 


.00 467 


.83 626 


37 


24 


.16 460 


.99 532 


.16928 


.83072 


.00 468 


.83 540 


36 


25 


9.16 545 


9.99 530 


9.17016 


0.82 984 


0.00 470 


0.83 455 


35 


26 


.16631 


.99 528 


.17 103 


.82 897 


.00 472 


.83 369 


34 


27 


.16716 


.99 526 


.17 190 


.82 810 


.00474 


.83284 


33 


28 


.16801 


.99524 


.17 277 


.82 723 


.00 476 


.83 199 


32 


29 


.16 886 


.99 522 


.17363 


.82 637 


.00 478 


.83114 


31 


30 


9.16 970 


9.99 520 


9.17 450 


0.82 550 


0.00 480 


0.83 030 


30 


31 


.17 055 


.99 518 


.17 536 


.82 464 


.00 482 


.82 945 


29 


32 


.17 139 


.99 517 


.17 622 


.82 378 


.00 483 


.82 861 


28 


33 


.17 223 


.99 515 


.17 708 


.82 292 


.00 485 


.82 777 


27 


34 


.17307 


.99 513 


.17 794 


.82 206 


.00 487 


.82 693 


26 


35 


9.17391 


9.99511 


9.17 880 


0.82 120 


0.00 489 


0.82 609 


25 


36 


.17474 


.99 509 


.17 965 


.82 035 


.00 491 


.82 526 


24 


37 


.17 558 


.99 507 


.18 051 


.81 949 


.00 493 


.82 442 


23 


38 


.17 641 


.99 505 


.18 136 


.81864 


.00 495 


.82 359 


22 


39 


.17 724 


.99 503 


.18221 


.81 779 


.00 497 


.82 276 


21 


40 


9.17 807 


9.99 501 


9.18 306 


0.81 694 


0.00 499 


0.82 193 


20 


41 


.17 890 


.99 499 


.18391 


.81 609 


.00 501 


.82 110 


19 


42 


.17 973 


.99 497 


.18 475 


.81 525 


.00 503 


.82 027 


18 


43 


.18 055 


.99 495 


.18 560 


.81 440 


.00505 


.81 945 


17 


44 


.18 137 


.99 494 


.18 644 


.81 356 


.00 506 


.81 863 


16 


45 


9.18 220 


9.99 492 


9.18728 


0.81 272 


0.00 508 


0.81 780 


15 


46 


.18 302 


.99 490 


.18812 


.81 188 


.00 510 


.81 698 


14 


47 


.18383 


.99 488 


.18 896 


.81 104 


.00 512 


.81 617 


13 


48 


.18465 


.99 486 


.18 979 


.81 021 


.00 514 


.81 535 


12 


49 


.18 547 


.99484 


.19 063 


.80 937 


.00 516 


.81 453 


11 


50 


9.18 628 


9.99 482 


9.19 146 


0.80 854 


0.00 518 


0.81 372 


10 


51 


.18 709 


.99 480 


.19 229 


.80 771 


.00520 


.81 291 


9 


52 


.18790 


.99 478 


.19312 


.80 688 


.00 522 


.81 210 


8 


53 


.18871 


.99 476 


.19 395 


.80 605 


.00 524 


.81 129 


7 


54 


.18 952 


.99 474 


.19 478 


.80522 


.00526 


.81 048 


6 


55 


9.19 033 


9.99 472 


9.19 561 


0.80 439 


0.00 528 


0.80 967 


5 


56 


.19 113 


.99 470 


.19 643 


.80 357 


.00 530 


.80 887 


4 


57 


.19 193 


.99 468 


.19 725 


.80 275 


.00532 


.80 807 


3 


58 


.19 273 


.99 466 


.19807 


.80 193 


.00 534 


.80 727 


2 


59 


.19 353 


.99464 


.19 889 


.80 111 


.00536 


.80 647 


1 


60 


9.19433 


9.99 462 


9.19971 


0.80 029 


0.00 538 


0.80 567 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



98 (278) 



(261) 81 



Table 4. Trigonometric Logarithms 



205 



9 (189) 



(350) 170 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







9.19 433 


9.99 462 


9.19971 


0.80 029 


0.00 538 


0.80 567 


60 


1 


.19513 


.99 460 


.20 053 


.79 947 


.00540 


.80 487 


59 


2 


.19 592 


.99 458 


.20 134 


.79 866 


.00542 


.80 408 


58 


3 


.19 672 


.99 456 


.20 216 


.79784 


.00544 


.80 328 


57 


4 


.19 751 


.99 454 


.20 297 


.79 703 


.00 546 


.80249 


56 


5 


9.19 830 


9.99 452 


9.20 378 


0.79 622 


0.00 548 


0.80 170 


55 


6 


.19 909 


.99 450 


.20 459 


.79541 


.00550 


.80 091 


54 


7 


.19 988 


.99 448 


.20 540 


.79 460 


.00552 


.80 012 


53 


8 


.20 067 


.99 446 


.20 621 


.79 379 


.00554 


.79 933 


52 


9 


.20 145 


.99444 


.20 701 


.79 299 


.00556 


.79 855 


51 


10 


9.20 223 


9.99 442 


9.20 782 


0.79 218 


0.00 558 


0.79 777 


50 


11 


.20 302 


.99 440 


.20 862 


.79 138 


.00560 


.79 698 


49 


12 


.20 380 


.99 438 


.20 942 


.79 058 


.00562 


.79 620 


48 


13 


.20 458 


.99436 


.21 022 


.78 978 


.00564 


.79 542 


47 


14 


.20 535 


.99 434 


.21 102 


.78 898 


.00566 


.79 465 


46 


15 


9.20 613 


9.99 432 


9.21 182 


0.78 818 


0.00 568 


0.79 387 


45 


16 


.20 691 


.99 429 


.21 261 


.78 739 


.00 571 


.79 309 


44 


17 


.20 768 


.99 427 


.21 341 


.78 659 


.00573 


.79 232 


43 


18 


.20845 


.99 425 


.21 420 


.78 580 


.00 575 


.79 155 


42 


19 


.20 922 


.99 423 


.21 499 


.78 501 


.00577 


.79 078 


41 


20 


9.20 999 


9.99 421 


9.21 578 


0.78 422 


0.00 579 


0.79 001 


40 


21 


.21 076 


.99 419 


.21 657 


.78 343 


.00581 


.78 924 


39 


22 


.21 153 


.99417 


.21 736 


.78 264 


.00583 


.78 847 


38 


23 


.21 229 


.99 415 


.21 814 


.78 186 


.00 585 


.78 771 


37 


24 


.21 306 


.99 413 


.21 893 


.78 107 


.00 587 


.78 694 


36 


25 


9.21 382 


9.99411 


9.21 971 


0.78 029 


0.00 589 


0.78 618 


35 


26 


.21 458 


.99 409 


.22 049 


.77 951 


.00 591 


.78 542 


34 


27 


.21534 


.99 407 


.22 127 


.77 873 


.00593 


-.78466 


33 


28 


.21 610 


.99404 


.22 205 


.77 795 


.00596 


.78 390 


32 


29 


.21 685 


.99 402 


.22 283 


.77 717 


.00598 


.78 315 


31 


30 


9.21 761 


9.99 400 


9.22 361 


0.77 639 


0.00 600 


0.78 239 


30 


31 


.21 836 


.99 398 


.22 438 


.77 562 


.00602 


.78164 


29 


32 


.21 912 


.99 396 


.22 516 


.77 484 


.00604 


.78 088 


28 


33 


.21 987 


.99394 


.22 593 


.77 407 


.00 606 


.78 013 


27 


34 


.22 062 


.99 392 


.22 670 


.77 330 


.00608 


.77 938 


26 


35 


9.22 137 


9.99 390 


9.22 747 


0.77 253 


0.00 610 


0.77 863 


25 


36 


.22211 


.99 388 


.22 824 


.77 176 


.00612 


.77 789 


24 


37 


.22 286 


.99 385 


.22 901 


.77 099 


.00615 


.77 714 


23 


38 


.22 361 


.99383 


.22 977 


.77 023 


.00 617 


.77 639 


22 


39 


.22 435 


.99 381 


.23054 


.76 946 


.00619 


.77 565 


21 


40 


9.22 509 


9.99 379 


9.23 130 


0.76 870 


0.00 621 


0.77 491 


20 


41 


.22 583 


.99 377 


.23 206 


.76 794 


.00 623 


.77 417 


19 


42 


.22 657 


.99 375 


.23 283 


.76 717 


.00625 


.77 343 


18 


43 


.22 731 


.99 372 


.23 359 


.76641 


.00628 


.77 269 


17 


44 


.22 805 


.99 370 


.23 435 


.76 565 


.00630 


.77 195 


16 


45 


9.22 878 


9.99 368 


9.23 510 


0.76 490 


0.00 632 


0.77 122 


15 


46 


.22 952 


.99 366 


.23 586 


.76 414 


.00634 


.77048 


14 


47 


.23 025 


.99364 


.23 661 


.76 339 


.00636 


.76 975 


13 


48 


.23 098 


.99 362 


.23 737 


.76 263 


.00638 


.76 902 


12 


49 


.23 171 


.99 359 


.23 812 


.76 188 


.00641 


.76 829 


11 


50 


9.23 244 


9.99 357 


9.23 887 


0.76 113 


0.00 643 


0.76 756 


10 


51 


.23 317 


.99 355 


.23 962 


.76 038 


.00645 


.76683 


9 


52 


.23 390 


.99 353 


.24 037 


.75 963 


.00647 


.76 610 


8 


53 


.23 462 


.99 351 


.24 112 


.75 888 


.00649 


.76 538 


7 


54 


.23 535 


.99 348 


.24 186 


.75 814 


.00652 


.76 465 


6 


55 


9.23 607 


9.99 346 


9.24 261 


0.75 739 


0.00654 


0.76 393 


5 


56 


.23 679 


.99 344 


.24 335 


.75 665 


.00656 


.76 321 


4 


57 


.23 752 


.99342 


.24 410 


.75 590 


.00 658 


.76 248 


3 


58 


.23 823 


.99 340 


.24484 


.75 516 


.00 660 


.76 177 


2 


59 


.23 895 


.99 337 


.24 558 


.75 442 


.00 663 


.76 105 


1 


60 


9.23 967 


9.99 335 


9.24 632 


0.75 368 


0.00 665 


0.76 033 





Cos 


Sin 


Cot 


Tail 


Csc 


Sec 


' 



99 (279) 



(260) 80 



206 



Table 4. Trigonometric Logarithms 



10 (190) 



(349) 169 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







9.23 967 


9.99 335 


9.24 632 


0.75 368 


0.00 665 


0.76 033 


60 


1 


.24 039 


.99 333 


.24 706 


.75 294 


.00 667 


.75 961 


59 


2 


.24 110 


.99 331 


.24 779 


.75 221 


.00 669 


.75 890 


58 


3 


.24 181 


.99 328 


.24 853 


.75 147 


.00 672 


.75 819 


57 


4 


.24 253 


.99 326 


.24 926 


.75 074 


.00 674 


.75 747 


56 


5 


9.24 324 


9.99 324 


9.25 000 


0.75 000 


0.00 676 


0.75 676 


55 


6 


.24 395 


.99 322 


.25 073 


.74 927 


.00 678 


.75 605 


54 


7 


.24 466 


.99 319 


.25 146 


.74 854 


.00 681 


.75 534 


53 


8 


.24 536 


.99317 


.25 219 


.74 781 


.00683 


.75 464 


52 


9 


.24 607 


.99 315 


.25 292 


.74 708 


.00 685 


.75 393 


51 


10 


9.24 677 


9.99 313 


9.25 365 


0.74 635 


0.00 687 


0.75 323 


50 


11 


.24 748 


.99 310 


.25 437 


.74 563 


.00 690 


.75 252 


49 


12 


.24 818 


.99 308 


.25 510 


.74 490 


.00692 


.75 182 


48 


13 


.24 888 


.99 306 


.25 582 


.74 418 


.00 694 


.75 112 


47 


14 


.24 958 


.99 304 


.25 655 


.74 345 


.00696 


.75 042 


46 


15 


9.25 028 


9.99 301 


9.25 727 


0.74 273 


0.00 699 


0.74 972 


45 


16 


.25 098 


.99 299 


.25 799 


.74 201 


.00 701 


.74 902 


44 


17 


.25 168 


.99 297 


.25 871 


.74 129 


.00 703 


.74 832 


43 


18 


.25 237 


.99 294 


.25 943 


.74 057 


.00 706 


.74 763 


42 


19 


.25 307 


.99 292 


.26 015 


.73 985 


.00 708 


.74 693 


41 


20 


9.25 376 


9.99 290 


9.26 086 


0.73 914 


0.00 710 


0.74 624 


40 


21 


.25 445 


.99 288 


.26 158 


.73842 


.00712 


.74 555 


39 


22 


.25 514 


.99 285 


.26 229 


.73 771 


.00 715 


.74 486 


38 


23 


.25 583 


.99 283 


.26 301 


.73 699 


.00 717 


.74 417 


37 


24 


.25 652 


.99 281 


.26 372 


.73 628 


.00719 


.74 348 


36 


25 


9.25 721 


9.99 278 


9.26 443 


0.73 557 


0.00 722 


0.74 279 


35 


26 


.25 790 


.99 276 


.26 514 


.73 486 


.00 724 


.74 210 


34 


27 


.25 858 


.99 274 


.26 585 


.73 415 


.00 726 


.74 142 


33 


28 


.25 927 


.99 271 


.26 655 


.73 345 


.00 729 


.74 073 


32 


29 


.25 995 


.99 269 


.26 726 


.73 274 


.00 731 


.74 005 


31 


30 


9.26 063 


9.99 267 


9.26 797 


0.73 203 


0.00 733 


0.73 937 


30 


31 


.26 131 


.99 264 


.26 867 


.73 133 


.00 736 


.73 869 


29 


32 


.26 199 


.99 262 


.26 937 


.73 063 


.00 738 


.73 801 


28 


33 


.26 267 


.99 260 


.27 008 


.72 992 


.00740 


.73 733 


27 


34 


.26 335 


.99 257 


.27 078 


.72 922 


.00 743 


.73 665 


26 


35 


9.26 403 


9.99 255 


9.27 148 


0.72 852 


0.00 745 


0.73 597 


25 


36 


.26 470 


.99 252 


.27 218 


.72 782 


.00 748 


.73 530 


24 


37 


.26 538 


.99 250 


.27 288 


.72 712 


.00 750 


.73 462 


23 


38 


.26 605 


.99 248 


.27 357 


.72 643 


.00 752 


.73 395 


22 


39 


.26 672 


.99 245 


.27 427 


.72 573 


.00 755 


.73 328 


21 


40 


9.26 739 


9.99 243 


9.27 496 


0.72 504 


0.00 757 


0.73 261 


20 


41 


.26 806 


.99 241 


.27 566 


.72 434 


.00 759 


.73 194 


19 


42 


.26 873 


.99 238 


.27 635 


.72 365 


.00 762 


.73 127 


18 


43 


.26 940 


.99 236 


.27 704 


.72 296 


.00 764 


.73 060 


17 


44 


.27 007 


.99 233 


.27 773 


.72 227 


.00 767 


.72 993 


16 


45 


9.27 073 


9.99 231 


9.27 842 


0.72 158 


0.00 769 


0.72 927 


15 


46 


.27 140 


.99 229 


.27911 


.72 089 


.00 771 


.*2 860 


14 


47 


.27 206 


.99 226 


.27 980 


.72 020 


.00 774 


.72 794 


13 


48 


.27 273 


.99 224 


.28 049 


.71 951 


.00776 


.72 727 


12 


49 


.27 339 


.99 221 


.28 117 


.71 883 


.00 779 


.72 661 


11 


50 


9.'27 405 


9.99 219 


9.28 186 


0.71 814 


0.00 781 


0.72 595 


10 


51 


.27 471 


.99 217 


.28 254 


.71 746 


.00 783 


.72 529 


9 


52 


.27 537 


.99 214 


.28 323 


.71 677 


.00 786 


.72 463 


8 


53 


.27 602 


.99 212 


.28 391 


.71 609 


.00 788 


.72 398 


7 


54 


.27 668 


.99 209 


.28 459 


.71 541 


.00 791 


.72 332 


6 


55 


9.27 734 


9.99 207 


9.28 527 


0.71 473 


0.00 793 


0.72 266 


5 


56 


.27 799 


.99 204 


.28 595 


.71 405 


.00 796 


.72 201 


4 


57 


.27 864 


.99 202 


.28 662 


.71 338 


.00 798 


.72 136 


3 


58 


.27 930 


.99 200 


.28 730 


.71 270 


.00 800 


.72 070 


2 


59 


.27 995 


.99 197 


.28 798 


.71 202 


.00 803 


.72 005 


1 


60 


9.28 060 


9.99 195 


9.28 865 


0.71 135 


0.00 805 


0.71 940 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



100 (280) 



(259) 79 



Table 4. Trigonometric Logarithms 



207 



11 (191) 



(348) 168 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







9.28 060 


9.99 195 


9.28 865 


0.71 135 


0.00 805 


0.71 940 


60 


1 


.28 125 


.99 192 


.28 933 


.71 067 


.00808 


.71 875 


59 


2 


.28 190 


.99 190 


.29 000 


.71 000 


.00810 


.71 810 


58 


3 


.28 254 


.99 187 


.29 067 


.70 933 


.00813 


.71 746 


57 


4 


.28 319 


.99185 


.29 134 


.70 866 


.00815 


.71 681 


56 


5 


9.28 384 


9.99 182 


9.29 201 


0.70 799 


0.00 818 


0.71 616 


55 


6 


.28448 


.99 180 


.29 268 


.70 732 


.00820 


.71 552 


54 


7 


.28 512 


.99 177 


.29 335 


.70 665 


.00823 


.71 488 


53 


8 


.28 577 


.99 175 


.29 402 


.70 598 


.00825 


.71 423 


52 


9 


.28641 


.99 172 


.29 468 


.70 532 


.00 828 


.71 359 


51 


10 


9.28 705 


9.99 170 


9.29 535 


0.70 465 


0.00 830 


0.71 295 


50 


11 


.28 769 


.99 167 


.29 601 


.70 399 


.00833 


.71 231 


49 


12 


.28833 


.99 165 


.29 668 


.70 332 


.00835 


.71 167 


48 


13 


.28 896 


.99 162 


.29 734 


.70 266 


.00838 


.71 104 


47 


14 


.28 960 


.99 160 


.29 800 


.70 200 


.00840 


.71040 


46 


15 


9.29 024 


9.99 157 


9.29 866 


0.70 134 


0.00 843 


0.70 976 


45 


16 


.29 087 


.99 155 


.29 932 


.70 068 


.00845 


.70 913 


44 


17 


.29 150 


.99 152 


.29 998 


.70 002 


.00848 


.70 850 


43 


18 


.29 214 


.99 150 


.30 064 


.69 936 


.00 850 


.70 786 


42 


19 


.29 277 


.99 147 


.30 130 


.69 870 


.00853 


.70 723 


41 


20 


9.29 340 


9.99 145 


9.30 195 


0.69 805 


0.00 855 


0.70 660 


40 


21 


.29 403 


.99 142 


.30 261 


.69 739 


.00858 


.70 597 


39 


22 


.29 466 


.99 140 


.30 326 


.69 674 


.00 860 


. .70534 


38 


23 


.29 529 


.99 137 


.30 391 


.69 609 


.00863 


.70 471 


37 


24 


.29 591 


.99 135 


.30 457 


.69543 


.00865 


.70 409 


36 


25 


9.29 654 


9.99 132 


9.30 522 


0.69 478 


0.00 868 


0.70 346 


35 


26 


.29 716 


.99 130 


.30 587 


.69 413 


.00870 


.70284 


34 


27 


.29 779 


.99 127 


.30 652 


.69 348 


.00873 


.70 221 


33 


28 


.29841 


.99 124 


.30 717 


.69 283 


.00876 


.70 159 


32 


29 


.29 903 


.99 122 


. .30 782 


.69 218 


.00878 


.70 097 


31 


30 


9.29 966 


9.99 119 


9.30 846 


0.69 154 


0.00 881 


0.70 034 


30 


31 


.30 028 


.99 fl7 


.30911 


.69 089 


.00883 


.69 972 


29 


32 


.30 090 


.99 114 


.30 975 


.69 025 


.00886 


.69 910 


28 


33 


.30 151 


.99 112 


.31 040 


.68 960 


.00888 


.69849 


27 


34 


.30 213 


.99 109 


.31 104 


.68 896 


.00 891 


.69 787 


26 


35 


9.30 275 


9.99 106 


9.31 168 


0.68 832 


0.00 894 


0.69 725 


25 


36 


.30 336 


.99 104 


.31 233 


.68 767 


.00 896 


.69 664 


24 


37 


.30 398 


.99 101 


.31 297 


.68 703 


.00 899 


.69 602 


23 


38 


.30 459 


.99 099 


.31 361 


.68 639 


.00 901 


.69 541 


22 


39 


.30 521 


.99 096 


.31 425 


.68 575 


.00904 


.69 479 


21 


40 


9.30 582 


9.99 093 


9.31 489 


0.68 511 


0.00 907 


0.69 418 


20 


41 


.30643 


.99 091 


.31 552 


.68 448 


.00909 


.69 357 


19 


42 


.30704 


.99088 


.31 616 


.68384 


.00 912 


.69 296 


18 


43 


.30 765 


.99 086 


.31 679 


.68 321 


.00914 


.69 235 


17 


44 


.30 826 


.99 083 


.31 743 


.68 257 


.00917 


.69 174 


16 


45 


9.30 887 


9.99 080 


9.31 806 


0.68 194 


0.00 920 


0.69 113 


15 


46 


.30 947 


.99 078 


.31 870 


.68 130 


.00922 


.69 053 


14 


47 


.31 008 


.99 075 


.31 933 


.68 067 


.00925 


.68 992 


13 


48 


.31 068 


.99 072 


.31 996 


.68004 


.00928 


.68 932 


12 


49 


.31 129 


.99 070 


.32 059 


.67 941 


.00930 


.68 871 


11 


50 


9.31 189 


9.99 067 


9.32 122 


0.67 878 


0.00 933 


0.68 811 


10 


51 


.31 250 


.99 064 


.32 185 


.67 815 


.00936 


.68 750 


9 


52 


.31 310 


.99 062 


.32 248 


.67 752 


.00938 


.68 690 


8 


53 


.31 370 


.99 059 


.32311 


.67 689 


.00941 


.68 630 


7 


54 


.31 430 


.99 056 


.32 373 


.67 627 


.00944 


.68570 


6 


55 


9.31 490 


9.99 054 


9.32 436 


0.67 564 


0.00 946 


0.68 510 


5 


56 


.31 549 


.99 051 


.32 498 


.67 502 


.00949 


.68 451 


4 


57 


.31 609 


.99048 


.32 561 


.67 439 


.00952 


.68 391 


3 


58 


.31 669 


.99046 


.32 623 


.67 377 


.00954 


.68 331 


2 


59 


.31 728 


.99043 


.32685 


.67 315 


.00957 


.68272 


1 


60 


9.31 788 


9.99 040 


9.32 747 


0.67 253 


0.00960 


0.68 212 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



101 (281) 



(258) 78 



208 



Table 4. Trigonometric Logarithms 



12 (192) 



(347) 167 c 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


CM- 







9.31 788 


9.99 040 


9.32 747 


0.67 253 


0.00 960 


0.68 212 


60 


1 


.31 847 


.99 038 


.32 810 


.67 190 


.00 962 


.68 153 


59 


2 


.31 907 


.99 035 


.32 872 


.67 128 


.00 965 


.68 093 


58 


3 


.31 966 


.99 032 


.32 933 


.67 067 


.00 968 


.68 034 


57 


4 


.32 025 


.99 030 


.32 995 


.67 005 


.00 970 


.67 975 


56 


5 


9.32 084 


9.99 027 


9.33 057 


0.66 943 


0.00 973 


0.67 916 


55 


6 


.32 143 


.99 024 


.33 119 


.66 881 


.00 976 


.67 857 


54 


7 


.32 202 


.99 022 


.33 180 


.66 820 


.00 978 


.67 798 


53 


8 


.32 261 


.99 019 


.33 242 


.66 758 


.00 981 


.67 739 


52 


9 


.32 319 


.99 016 


.33 303 


.66 697 


.00984 


.67 681 


51 


10 


9.32 378 


9.99 013 


9.33 365 


0.66 635 


0.00 987 


0.67 622 


50 


11 


.32 437 


.99 Oil 


.33 426 


.66 574 


.00 989 


.67 563 


49 


12 


.32 495 


.99 008 


.33 487 


.66 513 


.00 992 


.67 505 


48 


13 


.32 553 


.99 005 


.33 548 


.66 452 


.00 995 


.67 447 


47 


14 


.32 612 


.99 002 


.33 609 


.66 391 


.00 998 


.67 388 


46 


15 


9.32 670 


9.99 000 


9.33 670 


0.66 330 


0.01 000 


0.67 330 


45 


16 


.32 728 


.98 997 


.33 731 


.66 269 


.01 003 


.67 272 


44 


17 


.32 786 


.98 994 


.33 792 


.66 208 


.01 006 


.67 214 


43 


18 


.32844 


.98 991 


.33 853 


.66 147 


.01 009 


.67 156 


42 


19 


.32 902 


.98 989 


.33 913 


.66 087 


.01 Oil 


.67 098 


41 


20 


9.32 960 


9.98 986 


9.33 974 


0.66 026 


0.01 014 


0.67 040 


40 


21 


.33 018 


.98 983 


.34 034 


.65 966 


.01 017 


.66 982 


39 


22 


.33 075 


.98 980 


.34 095 


.65 905 


.01 020 


.66 925 


38 


23 


.33 133 


.98 978 


.34 155 


.65845 


.01 022 


.66 867 


37 


24 


.33 190 


.98 975 


.34 215 


.65 785 


.01 025 


.66 810 


36 


25 


9.33 248 


9.98 972 


9.34 276 


0.65 724 


0.01 028 


0.66 752 


35 


26 


.33 305 


.98969 


.34 336 


.65 664 


.01 031 


.66 695 


34 


27 


.33 362 


.98 967 


.34 396 


.65604 


.01 033 


.66 638 


33 


28 


.33 420 


.98 964 


.34 456 


.65544 


.01 036 


.66 580 


32 


29 


.33 477 


.98 961 


.34 516 


.65484 


.01 039 


.66 523 


31 


30 


9.33 534 


9.98 958 


9.34 576 


0.65 424 


0.01 042 


0.66 466 


30 


31 


.33 591 


.98 955 


.34 635 


.65 365 


*.01 045 


.66 409 


29 


32 


.33 647 


.98 953 


.34 695 


.65 305 


.01 047 


.66 353 


28 


33 


.33 704 


.98 950 


.34 755 


.65 245 


.01 050 


.66 296 


27 


34 


.33 761 


.98 947 


.34 814 


.65 186 


.01 053 


.66 239 


26 


35 


9.33 818 


9.98 944 


9.34 874 


0.65 126 


0.01 056 


0.66 182 


25 


36 


.33 874 


.98 941 


.34 933 


.65 067 


.01 059 


.66 126 


24 


37 


.33 931 


.98 938 


.34 992 


.65 008 


.01 062 


.66 069 


23 


38 


.33 987 


.98 936 


.35 051 


.64 949 


.01 064 


.66 013 


22 


39 


.34 043 


.98 933 


.35 111 


.64 889 


.01 067 


.65 957 


21 


40 


9.34 100 


9.98 930 


9.35 170 


0.64 830 


0.01 070 


0.65 900 


20 


41 


.34 156 


.98 927 


.35 229 


.64771 


.01 073 


.65 844 


19 


42 


.34 212 


.98 924 


.35 288 


.64712 


.01 076 


.65 788 


18 


43 


.34 268 


.98 921 


.35 347 


.64 653 


.01 079 


.65 732 


17 


44 


.34 324 


.98 919 


.35 405 


.64595 


.01 081 


.65 676 


16 


45 


9.34 380 


9.98 916 


9.35 464 


0.64 536 


0.01 084 


0.65 620 


15 


46 


.34 436 


.98 913 


.35 523 


.64477 


.01 087 


.65 564 


14 


47 


.34 491 


.98 910 


.35 581 


.64419 


.01 090 


65509 


13 


48 


.34 547 


.98 907 


.35 640 


.64360 


.01 093 


.65 453 


12 


49 


.34 602 


.98 904 


.35 698 


.64302 


.01 096 


.65 398 


11 


50 


9.34 658 


9.98 901 


9.35 757 


0.64 243 


0.01 099 


0.65 342 


10 


51 


.34 713 


.98 898 


.35 815 


.64185 


.01 102 


.65 287 


9 


52 


.34 769 


.98 896 


.35 873 


.64 127 


.01 104 


.65 231 


8 


53 


.34 824 


.98 893 


.35 931 


.64069 


.01 107 


.65 176 


7 


54 


.34 879 


.98 890 


.35 989 


.64011 


.01 110 


.65 121 


6 


55 


9.34 934 


9.98 887 


9.36 047 


0.63 953 


0.01 113 


0.65 066 


5 


56 


.34 989 


.98884 


.36 105 


.63 895 


.01 116 


.65011 


4 


57 


.35 044 


.98 881 


.36 163 


.63 837 


.01 119 


.64956 


3 


58 


.35 099 


.98 878 


.36 221 


.63 779 


.01 122 


.64 901 


2 


59 


.35 154 


.98 875 


.36 279 


.63 721 


.01 125 


.64 846 


1 


60 


9.35 209 


9.98 872 


9.36 336 


0.63 664 


0.01 128 


0.64 791 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



102 (282) 



(257) 77 C 



Table 4. Trigonometric Logarithms 



209 



13 (193) 



(346) 166 



' 


Sin 


Cos 


Tan Cot 


Sec 


Csc 







9.35 209 


9.98 872 


9.36 336 


0.63 664 


0.01 128 


0.64 791 


60 


1 


.35 263 


.98 869 


.36 394 


.63 606 


.01 131 


.64737 


59 


2 


.35 318 


.98 867 


.36 452 


.63 548 


.01 133 


.64 682 


58 


3 


.35 373 


.98864 


.36 509 


.63 491 


.01 136 


.64 627 


57 


4 


.35 427 


.98 861 


.36 566 


.63 434 


.01 139 


.64 573 


56 


5 


9.35 481 


9.98 858 


9.36 624 


0.63 376 


0.01 142 


0.64 519 


55 


6 


.35 536 


.98855 


.36 681 


.63 319 


.01 145 


.64464 


54 


7 


.35 590 


.98852 


.36 738 


.63 262 


.01 148 


.64410 


53 


8 


.35 644 


.98849 


.36 795 


.63 205 


.01 151 


.64356 


52 


9 


.35 698 


.98846 


.36 852 


.63 148 


.01 154 


.64302 


51 


10 


9.35 752 


9.98 843 


9.36 909 


0.63 091 


0.01 157 


0.64 248 


50 


11 


.35 806 


.98840 


.36 966 


.63 034 


.01 160 


.64 194 


49 


12 


.35 860 


.98 837 


.37 023 


.62 977 


.01 163 


.64 140 


48 


13 


.35 914 


.98 834 


.37 080 


.62 920 


.01 166 


.64 086 


47 


14 


.35 968 


.98 831 


.37 137 


.62 863 


.01 169 


.64032 


46 


15 


9.36 022 


9.98 828 


9.37 193 


0.62 807 


0.01 172 


0.63 978 


45 


16 


.36 075 


.98 825 


.37 250 


.62 750 


.01 175 


.63 925 


44 


17 


.36 129 


.98 822 


.37 306 


.62 694 


.01 178 


.63 871 


43 


18 


.36 182 


.98 819 


.37 363 


.62 637 


.01 181 


.63 818 


42 


19 


.36 236 


.98 816 


.37 419 


.62 581 


.01 184 


.63 764 


41 


20 


9.36 289 


9.98 813 


9.37 476 


0.62 524 


0.01 187 


0.63 711 


40 


21 


.36 342 


.98 810 


.37 532 


.62 468 


.01 190 


.63 658 


39 


22 


.36 395 


.98 807 


.37 588 


.62 412 


.01 193 


.63 605 


38 


23 


.36 449 


.98804 


.37644 


.62 356 


.01 196 


.63 551 


37 


24 


.36 502 


.98 801 


.37 700 


.62 300 


.01 199 


.63 498 


36 


25 


9.36 555 


9.98 798 


9.37 756 


0.62 244 


0.01 202 


0.63 445 


35 


26 


.36 608 


.98 795 


.37 812 


.62 188 


.01 205 


.63 392 


34 


27 


.36 660 


.98 792 


.37 868 


.62 132 


.01 208 


.63 340 


33 


28 


.36 713 


.98 789 


.37 924 


.62 076 


.01 211 


.63 287 


32 


29 


.36 766 


.98 786 


.37980 


.62 020 


.01 214 


.63 234 


31 


30 


9.36 819 


9.98 783 


9.38 035 


0.61 965 


0.01 217 


0.63 181 


30 


31 


.36 871 


.98 780 


.38 091 


.61 909 


.01 220 


.63 129 


29 


32 


.36 924 


.98 777 


.38 147 


.61 853 


.01 223 


.63 076 


28 


33 


.36 976 


.98 774 


.38202 


.61 798 


.01 226 


.63 024 


27 


34 


.37 028 


.98 771 


.38 257 


.61 743 


.01 229 


.62 972 


26 


35 


9.37 081 


9.98 768 


9.38 313 


0.61 687 


0.01 232 


0.62 919 


25 


36 


.37 133 


.98 765 


.38368 


.61 632 


.01 235 


.62 867 


24 


37 


.37 185 


.98 762 


.38 423 


.61 577 


.01 238 


.62 815 


23 


38 


.37 237 


.98 759 


.38 479 


.61 521 


.01 241 


.62 763 


22 


39 


.37 289 


.98 756 


.38 534 


.61 466 


.01 244 


.62711 


21 


40 


9.37 341 


9.98 753 


9.38 589 


0.61 411 


0.01 247 


0.62 659 


20 


41 


.37 393 


.98 750 


.38644 


.61 356 


.01 250 


.62 607 


19 


42 


.37445 


.98 746 


.38 699 


.61 301 


.01 254 


.62 555 


18 


43 


.37 497 


.98 743 


.38754 


.61 246 


.01 257 


.62 503 


17 


44 


.37 549 


.98 740 


.38 808 


.61 192 


.01 260 


.62 451 


16 


45 


9.37 600 


9.98 737 


9.38 863 


0.61 137 


0.01 263 


0.62 400 


15 


46 


.37 652 


.98 734 


.38 918 


.61 082 


.01 266 


.62 348 


14 


47 


.37 703 


.98 731 


.38 972 


.61 028 


.01 269 


.62 297 


13 


48 


.37 755 


.98 728 


.39 027 


.60 973 


.01 272 


.62 245 


12 


49 


.37806 


.98 725 


.39 082 


.60 918 


.01 275 


.62 194 


11 


50 


9.37 858 


9.98 722 


9.39 136 


0.60 864 


0.01 278 


0.62 142 


10 


51 


.37909 


.98 719 


.39 190 


.60 810 


.01 281 


.62 091 


9 


52 


.37 960 


.98 715 


.39 245 


.60 755 


.01 285 


.62040 


8 


53 


.38011 


.98 712 


.39 299 


.60 701 


.01 288 


.61 989 


7 


54 


.38 062 


.98 709 


.39 353 


.60647 


.01 291 


.61 938 


6 


55 


9.38 113 


9.98 706 


9.39 407 


0.60 593 


0.01 294 


0.61 887 


5 


56 


.38 164 


.98 703 


.39 461 


.60 539 


.01 297 


.61836 


4 


57 


.38 215 


.98700 


.39 515 


.60 485 


.01 300 


.61 785 


3 


58 


.38 266 


.98 697 


.39 569 


.60431 


.01 303 


.61 734 


2 


59 


.38 317 


.98 694 


.39 623 


.60 377 


.01 306 


.61683 


1 


60 


9.38 368 


9.98 690 


9.39 677 


0.60 323 


0.01 310 


0.61 632 







Cos Sin 


Cot 


Tan 


Csc 


GAA 


' 



103 (283) 



(256) 76 



210 



Table 4. Trigonometric Logarithms 



14 (194) 



(345) 165 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







9.38 368 


9.98 690 


9.39 677 


0.60 323 


0.01 310 


0.61 632 


60 


1 


.38418 


.98 687 


.39 731 


.60 269 


.01 313 


.61 582 


59 


2 


.38 469 


.98 684 


.39 785 


.60 215 


.01 316 


.61 531 


58 


3 


.38519 


.98 681 


.39 838 


.60 162 


.01 319 


.61 481 


57 


4 


.38 570 


.98 678 


.39 892 


.60 108 


.01 322 


.61 430 


56 


5 


9.38 620 


9.98 675 


9.39 945 


0.60 055 


0.01 325 


0.61 380 


55 


6 


.38 670 


.98 671 


.39 999 


.60 001 


.01 329 


.61 330 


54 


7 


.38 721 


.98 668 


.40 052 


.59 948 


.01 332 


.61 279 


53 


8 


.38 771 


.98 665 


.40 106 


.59 894 


.01 335 


.61 229 


52 


9 


.38 821 


.98 662 


.40 159 


.59841 


.01 338 


.61 179 


51 


10 


9.38 871 


9.98 659 


9.40 212 


0.59 788 


0.01 341 


0.61 129 


50 


11 


.38 921 


.98 656 


.40 266 


.59 734 


.01 344 


.61 079 


49 


12 


.38 971 


.98 652 


.40 319 


.59 681 


.01 348 


.61 029 


48 


13 


.39 021 


.98 649 


.40 372 


.59 628 


.01 351 


.60 979 


47 


14 


.39 071 


.98 646 


.40 425 


.59 575 


.01 354 


.60 929 


46 


15 


9.39 121 


9.98 643 


9.40 478 


0.59 522 


0.01 357 


0.60 879 


45 


16 


.39 170 


.98 640 


.40 531 


.59 469 


.01 360 


.60 830 


44 


17 


.39 220 


.98 636 


.40 584 


.59 416 


.01 364 


.60 780 


43 


18 


.39 270 


.98 633 


.40 636 


.59364 


.01 367 


.60 730 


42 


19 


.39 319 


.98 630 


.40 689 


.59 311 


.01 370 


.60 681 


41 


20 


9.39 369 


9.98 627 


9.40 742 


0.59 258 


0.01 373 


0.60 631 


40 


21 


.39 418 


.98 623 


.40 795 


.59 205 


.01 377 


.60 582 


39 


22 


.39 467 


.98 620 


.40 847 


.59 153 


.01 380 


.60 533 


38 


23 


.39 517 


.98 617 


.40 900 


.59 100 


.01 383 


.60 483 


37 


24 


.39 566 


.98 614 


.40 952 


.59 048 


.01 386 


.60 434 


36 


25 


9.39 615 


9.98610 


9.41 005 


0.58 995 


0.01 390 


0.60 385 


35 


26 


.39 664 


.98 607 


.41 057 


.58 943 


.01 393 


.60 336 


34 


27 


.39 713 


.98 604 


.41 109 


.58 891 


.01 396 


.60 287 


33 


28 


.39 762 


.98 601 


.41 161 


.58 839 


.01 399 


.60 238 


32 


29 


.39 811 


.98 597 


.41 214 


.58 786 


.01 403 


.60 189 


31 


30 


9.39 860 


9.98 594 


9.41 266 


0.58 734 


0.01 406 


0.60 140 


30 


31 


.39 909 


.98 591 


.41 318 


.58 682 


.01 409 


.60 091 


29 


32 


.39 958 


.98 588 


.41 370 


.58 630 


.01 412 


.60 042 


28 


33 


.40 006 


.98 584 


.41 422 


.58 578 


.01 416 


.59 994 


27 


34 


.40 055 


.98 581 


.41 474 


.58 526 


.01 419 


.59 945 


26 


35 


9.40 103 


9.98 578 


9.41 526 


0.58 474 


0.01 422 


0.59 897 


25 


36 


.40 152 


.98 574 


.41 578 


.58 422 


.01 426 


.59848 


24 


37 


.40 200 


.98 571 


.41 629 


.58 371 


.01 429 


.59 800 


23 


38 


.40 249 


.98 568 


.41 681 


.58319 


.01 432 


.59 751 


22 


39 


.40 297 


.98 565 


.41 733 


.58 267 


.01 435 


.59 703 


21 


40 


9.40 346 


9.98 561 


9.41 784 


0.58 216 


0.01 439 


59 654 


20 


41 


.40 394 


.98 558 


.41 836 


.58 164 


.01 442 


.59 606 


19 


42 


.40 442' 


.98 555 


.41 887 


.58 113 


.01 445 


.59 558 


18 


43 


.40 490 


.98 551 


.41 939 


.58 061 


.01 449 


.59 510 


17 


44 


.40 538 


.98 548 


.41 990 


.58 010 


.01 452 


.59 462 


16 


45 


9.40 586 


9.98 545 


9.42 041 


0.57 959 


0.01 455 


0.59 414 


15 


46 


.40 634 


.98 541 


.42 093 


.57 907 


.01 459 


.59 366 


14 


47 


.40 682 


.98 538 


.42 144 


.57 856 


.01 462 


,69318 


13 


48 


.40 730 


.98 535 


.42 195 


.57 805 


.01465 


.59 270 


12 


49 


.40 778 


.98 531 


.42 246 


.57 754 


.01 469 


.59 222 


11 


50 


9.40 825 


9.98 528 


9.42 297 


0.57 703 


0.01 472 


0.59 175 


10 


51 


.40 873 


.98 525 


.42 348 


.57 652 


.01 475 


.59 127 


9 


52 


.40 921 


.98 521 


.42 399 


.57 601 


.01 479 


.59 079 


8 


53 


.40 968 


.98518 


.42 450 


.57 550 


.01 482 


.59 032 


7 


54 


,41 016 


.98 515 


.42 501 


.57 499 


.01 485 


.58 984 


6 


55 


9.41 063 


9.98511 


9.42 552 


0.57 448 


0.01 489 


0.58 937 


5 


56 


.41 111 


.98 508 


.42 603 


.57 397 


.01 492 


.58 889 


4 


57 


.41 158 


.98 505 


.42 653 


.57 347 


.01 495 


.58 842 


3 


58 


.41 205 


.98 501 


.42 704 


.57 296 


.01 499 


.58 795 


2 


59 


.41 252 


.98 498 


.42 755 


.57 245 


.01 502 


.58 748 


1 


60 


9.41 300 


9.98 494 


9.42 805 


0.57 195 


0.01 506 


0.58 700 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



104 (284) 



(255) 75 



Table 4. Trigonometric Logarithms 



211 



15 (195) 



(344) 164 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







9.41 300 


9.98 494 


9.42 805 


0.57 195 


0.01 506 


0.58 700 


60 


1 


.41 347 


.98 491 


.42856 


.57 144 


.01 509 


.58 653 


59 


2 


.41 394 


.US 4SS 


.42906 


.57 094 


.01 512 


.58606 


58 


3 


.41 441 


.98484 


.42 957 


.57043 


.01 516 


.58 559 


57 


4 


.41488 


.98 481 


.43007 


.56 993 


.01 519 


.58 512 


56 


5 


9.41 535 


9.98 477 


9.43 057 


0.56 943 


0.01 523 


0.58 465 


55 


6 


.41 582 


.98 474 


.43 108 


.56 892 


.01 526 


.58418 


54 


7 


.41 628 


.98 471 


.43 158 


.56842 


.01 529 


.58 372 


53 


8 


.41 675 


.98 467 


.43 208 


.56 792 


.01 533 


.58 325 


52 


9 


.41 722 


.98464 


.43 258 


.56 742 


.01 536 


.58 278 


51 


10 


9.41 768 


9.98 460 


9.43 308 


0.56 692 


0.01 540 


0.58 232 


50 


11 


.41 815 


.98 457 


.43 358 


.56 642 


.01543 


.58 185 


49 


12 


.41 861 


.98 453 


.43 408 


.56 592 


.01 547 


.58 139 


48 


13 


.41 908 


.98 450 


.43 458 


.56 542 


.01 550 


.58 092 


47 


14 


.41 954 


.98447 


.43 508 


.56 492 


.01 553 


.58 046 


46 


15 


9.42 001 


9.98 443 


9.43 558 


0.56 442 


0.01 557 


0.57 999 


45 


16 


.42047 


.98 440 


.43 607 


.56 393 


.01 560 


.57 953 


44 


17 


.42 093 


.98 436 


.43 657 


.56 343 


.01 564 


.57 907 


43 


18 


.42 140 


.98 433 


.43 707 


.56 293 


.01 567 


.57 860 


42 


19 


.42 186 


.98 429 


.43 756 


.56244 


.01 571 


.57 814 


41 


20 


9.42 232 


9.98 426 


9.43 806 


0.56 194 


0.01 574 


0.57 768 


40 


21 


.42 278 


.98 422 


.43855 


.56 145 


.01 578 


.57 722 


39 


22 


.42 324 


.98 419 


.43 905 


.56 095 


.01 581 


.57 676 


38 


23 


.42 370 


.98 415 


.43 954 


.56046 


.01 585 


.57 630 


37 


24 


.42 416 


.98 412 


.44004 


.55 996 


.01 588 


.57584 


36 


25 


9.42 461 


9.98 409 


9.44 053 


0.55 947 


0.01 591 


0.57 539 


35 


26 


.42 507 


.98 405 


.44 102 


.55 898 


.01 595 


.57 493 


34 


27 


.42 553 


.98 402 


.44 151 


.55849 


.01 598 


.57 447 


33 


28 


.42 599 


.98 398 


.44201 


.55 799 


.01 602 


.57 401 


32 


29 


.42644 


.98 395 


.44250 


.55 750 


.01 605 


.57 356 


31 


30 


9.42 690 


9.98 391 


9.44 299 


0.55 701 


0.01 609 


0.57 310 


30 


31 


.42 735 


.98 388 


.44 348 


.55 652 


.01 612 


.57 265 


29 


32 


.42 781 


.98384 


.44397 


.55 603 


.01 616 


.57 219 


28 


33 


.42 826 


.98 381 


.44446 


.55554 


.01 619 


.57 174 


27 


34 


.42 872 


.98 377 


.44 495 


.55 505 


.01 623 


.57 128 


26 


35 


9.42 917 


9.98 373 


9.44 544 


0.55 456 


0.01 627 


0.57 083 


25 


36 


.42 962 


.98 370 


.44592 


.55 408 


.01 630 


.57 038 


24 


37 


.43008 


.98 366 


.44641 


.55 359 


.01 634 


.56 992 


23 


38 


.43 053 


.98 363 


.44690 


.55 310 


.01 637 


.56 947 


22 


39 


.43 098 


.98 359 


.44738 


.55 262 


.01 641 


.56 902 


21 


40 


9.43 143 


9.98 356 


9.44 787 


0.55 213 


0.01 644 


0.56 857 


20 


41 


.43 188 


.98 352 


.44836 


.55 164 


.01648 


.56 812 


19 


42 


.43 233 


.98 349 


.44884 


.55 116 


.01 651 


.56 767 


18 


43 


.43 278 


.98 345 


.44933 


.55 067 


.01 655 


.56 722 


17 


44 


.43 323 


.98 342 


.44981 


.55 019 


.01 658 


.56 677 


16 


45 


9.43 367 


9.98 338 


9.45 029 


0.54 971 


0.01 662 


0.56 633 


15 


46 


.43 412 


.98 334 


.45 078 


.54922 


.01 666 


.56588 


14 


47 


.43 457 


.98 331 


.45 126 


.54874 


.01 669 


.56 543 


13 


48 


.43 502 


.98 327 


.45 174 


.54826 


.01 673 


.56 498 


12 


49 


.43546 


.98 324 


.45 222 


.54778 


.01 676 


.56454 


11 


50 


9.43 591 


9.98 320 


9.45 271 


0.54 729 


0.01 680 


0.56 409 


10 


51 


.43 635 


.98 317 


.45 319 


.54 681 


.01 683 


.56 365 


9 


52 


.43 680 


.98 313 


.45 367 


.54 633 


.01 687 


.56 320 


8 


53 


.43 724 


.98 309 


.45 415 


.54 585 


.01 691 


.56 276 


7 


54 


.43 769 


.98 306 


.45 463 


.54537 


.01 694 


.56 231 


6 


55 


9.43 813 


9.98 302 


9.45511 


0.54 489 


0.01 698 


0.56 187 


5 


56 


.43 857 


.98 299 


.45 559 


.54441 


.01 701 


.56 143 


4 


57 


.43 901 


.98 295 


.45606 


.54 394 


.01 705 


.56 099 


3 


58 


.43946 


.98 291 


.45654 


.54346 


.01 709 


.56 054 


2 


59 


.43 990 


.98 288 


.45 702 


.54298 


.01 712 


.56 010 


1 


60 


9.44 034 


9.98 284 


9.45 750 


0.54 250 


0.01 716 


0.55 966 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



105 (285) 



(254) 74 



212 



Table 4. Trigonometric Logarithms 



16 (196) 



(343) 163 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







9.44 034 


9.98 284 


9.45 750 


0.54 250 


0.01 716 


0.55 966 


60 


1 


.44 078 


.98 281 


.45 797 


.54 203 


.01 719 


.55 922 


59 


2 


.44 122 


.98 277 


.45845 


.54 155 


.01 723 


.55 878 


58 


3 


.44 166 


.98 273 


.45 892 


.54 108 


.01 727 


.55 834 


57 


4 


.44 210 


.98 270 


.45 940 


.54 060 


.01 730 


.55 790 


56 


5 


9.44 253 


9.98 266 


9.45 987 


0.54 013 


0.01 734 


0.55 747 


55 


6 


.44 297 


.98 262 


.46 035 


.53 965 


.01 738 


.55 703 


54 


7 


.44 341 


.98 259 


.46 082 


.53 918 


.01 741 


.55 659 


53 


8 


.44 385 


.98 255 


.46 130 


.53 870 


.01 745 


.55 615 


52 


9 


.44 428 


.98 251 


.46 177* 


.53 823 


.01 749 


.55 572 


51 


10 


9.44 472 


9.98 248 


9.46 224 


0.53 776 


0.01 752 


0.55 528 


50 


11 


.44516 


.98 244 


.46 271 


.53 729 


.01 756 


.55484 


49 


12 


.44 559 


.98 240 


.46 319 


.53 681 


.01 760 


.55 441 


48 


13 


.44 602 


.98 237 


.46 366 


.53 634 


.01 763 


.55 398 


47 


14 


.44 646 


.98 233 


.46 413 


.53 587 


.01 767 


.55 354 


46 


15 


9.44 689 


9.98 229 


9.46 460 


0.53 540 


0.01 771 


0.55311 


45 


16 


.44 733 


.98 226 


.46 507 


.53 493 


.01 774 


.55 267 


44 


17 


.44 776 


.98 222 


.46 554 


.53 446 


.01 778 


.55 224 


43 


18 


.44 819 


.98 218 


.46 601 


.53 399 


.01 782 


.55 181 


42 


19 


.44 862 


.98 215 


.46 648 


.53 352 


.01 785 


.55 138 


41 


20 


9.44 905 


9.98211 


9.46 694 


0.53 306 


0.01 789 


0.55 095 


40 


21 


.44 948 


.98 207 


.46 741 


.53 259 


.01 793 


.55 052 


39 


22 


.44 992 


.98 204 


.46 788 


.53 212 


.01 796 


.55 008 


38 


23 


.45 035 


.98 200 


.46 835 


.53 165 


.01 800 


.54965 


37 


24 


.45 077 


.98 196 


.46 881 


.53 119 


.01 804 


.54 923 


36 


25 


9.45 120 


9.98 192 


9.46 928 


0.53 072 


0.01 808 


0.54 880 


35 


26 


.45 163 


.98 189 


.46 975 


.53 025 


.01811 


.54837 


34 


27 


.45 206 


.98 185 


.47 021 


.52 979 


.01 815 


.54 794 


33 


28 


.45 249 


.98 181 


.47 068 


.52 932 


.01 819 


.54 751 


32 


29 


.45 292 


.98 177 


.47 114 


.52 886 


.01 823 


.54 708 


31 


30 


9.45 334 


9.98 174 


9.47 160 


0.52 840 


0.01 826 


0.54 666 


30 


31 


.45 377 


.98 170 


.47 207 


.52 793 


.01 830 


.54 623 


29 


32 


.45 419 


.98 166 


.47 253 


.52 747 


.01 834 


.54 581 


28 


33 


.45 462 


.98 162 


.47 299 


.52 701 


.01 838 


.54 538 


27 


34 


.45 504 


.98 159 


.47 346 


.52 654 


.01 841 


.54 496 


26 


35 


9.45 547 


9.98 155 


9.47 392 


0.52 608 


0.01 845 


0.54 453 


25 


36 


.45 589 


.98 151 


.47 438 


.52 562 


.01 849 


.54411 


24 


37 


.45 632 


.98 147 


.47484 


.52 516 


.01 853 


.54 368 


23 


38 


.45 674 


.98 144 


.47 530 


.52 470 


.01 856 


.54 326 


22 


39 


.45 716 


.98 140 


.47 576 


.52 424 


.01 860 


.54 284 


21 


40 


9.45 758 


9.98 136 


9.47 622 


0.52 378 


0.01 864 


0.54 242 


20 


41 


.45 801 


.98 132 


.47-668 


.52 332 


.01 868 


.54 199 


19 


42 


.45843 


.98 129 


.47 714 


.52 286 


.01 871 


.54 157 


18 


43 


.45 885 


.98 125 


.47 760 


.52 240 


.01 875 


.54 115 


17 


44 


.45 927 


.98 121 


.47 806 


.52 194 


.01 879 


.54 073 


16 


45 


9.45 969 


9.98 117 


9.47 852 


0.52 148 


0.01 883 


0.54 031 


15 


46 


.46011 


.98 113 


.47 897 


.52 103 


.01 887 


.53 989 


14 


47 


.46 053 


.98 110 


.47 943 


.52 057 


.01 890 


.53 947 


13 


48 


.46 095 


.98 106 


.47 989 


.52011 


.01 894 


.53 905 


12 


49 


.46 136 


.98 102 


.48 035 


.51 965 


.01 898 


.53 864 


11 


50 


9.46 178 


9.98 098 


9.48 080 


0.51 920 


0.01 902 


0.53 822 


10 


51 


.46 220 


.98 094 


.48 126 


.51 874 


.01 906 


.53 780 


9 


52 


.46 262 


.98 090 


.48 171 


.51 829 


.01 910 


.53 738 


8 


53 


.46 303 


.98 087 


.48 217 


.51 783 


.01 913 


.53 697 


7 


54 


.46 345 


.98 083 


.48 262 


.51 738 


.01 917 


.53 655 


6 


55 


9.46 386 


9.98079 


9.48 307 


0.51 693 


0.01 921 


0.53 614 


5 


56 


.46 428 


.98 075 


.48 353 


.51 647 


.01 925 


.53 572 


4 


57 


.46 469 


.98 07i 


.48 398 


.51 602 


.01 929 


.53 531 


3 


58 


.46511 


.98 067 


.48 443 


.51 557 


.01 933 


.53 489 


2 


59 


.46 552 


.98 063 


.48 489 


.51511 


.01 937 


.53 448 


1 


60 


9.46 594 


9.98 060 


9.48 534 


0.51 466 


0.01 940 


0.53 406 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



106 (286) 



(253) 73 



Table 4. Trigonometric Logarithms 



213 



17 (197) 



(342) 162 C 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







9.46 594 


9.98 060 


9.48 534 


0.51 466 


0.01 940 


0.53 406 


60 


1 


.46 635 


.98 056 


.48 579 


.51 421 


.01 944 


.53 365 


59 


2 


.46 676 


.98 052 


.48 624 


.51 376 


.01 948 


.53 324 


58 


3 


.46 717 


.98 048 


.48 669 


.51 331 


.01 952 


.53283 


57 


4 


.46 758 


.98044 


.48 714 


.51 286 


.01 956 


.53 242 


56 


5 


9.46 800 


9.98 040 


9.48 759 


0.51 241 


0.01 960 


0.53 200 


55 


6 


.46841 


.98 036 


.48804 


.51 196 


.01 964 


.53 159 


54 


7 


.46 882 


.98 032 


.48849 


.51 151 


.01 968 


.53 118 


53 


8 


.46 923 


.98 029 


.48 894 


.51 106 


.01 971 


.53 077 


52 


9 


.46964 


.98 025 


.48 939 


.51 061 


.01 975 


.53 036 


51 


10 


9.47 005 


9.98 021 


9.48 984 


0.51 016 


0.01 979 


0.52 995 


50 


11 


.47 045 


.98 017 


.49 029 


.50 971 


.01983 


.52 955 


49 


12 


.47 086 


.98 013 


.49 073 


.50 927 


.01 987 


.52 914 


48 


13 


.47 127 


.98 009 


.49 118 


.50 882 


.01 991 


.52 873 


47 


14 


.47 168 


.98005 


.49 163 


.50837 


.01 995 


.52832 


46 


15 


9.47 209 


9.98 001 


9.49 207 


0.50 793 


0.01 999 


0.52 791 


45 


16 


.47 249 


.97 997 


.49 252 


.50 748 


.02 003 


.52 751 


44 


17 


.47 290 


.97 993 


.49 296 


.50704 


.02 007 


.52 710 


43 


18 


.47 330 


.97 989 


.49 341 


.50 659 


.02011 


.52 670 


42 


19 


.47 371 


.97 986 


.49 385 


.50 615 


.02 014 


.52 629 


41 


20 


9.47411 


9.97 982 


9.49 430 


0.50 570 


0.02 018 


0.52 589 


40 


21 


.47 452 


.97 978 


.49 474 


.50 526 


.02 022 


.52548 


39 


22 


.47 492 


.97 974 


.49 519 


.50 481 


.02 026 


.52 508 


38 


23 


.47 533 


.97 970 


.49 563 


.50 437 


.02 030 


.52 467 


37 


24 


.47 573 


.97 966 


.49 607 


.50 393 


.02 034 


.52 427 


36 


25 


9.47 613 


9.97 962 


9.49 652 


0.50 348 


0.02 038 


0.52 387 


35 


26 


.47 654 


.97 958 


.49 696 


.50304 


.02042 


.52 346 


34 


27 


.47 694 


.97 954 


.49 740 


.50 260 


.02 046 


.52 306 


33 


28 


.47 734 


.97 950 


.49784 


.50 216 


.02 050 


.52 266 


32 


29 


.47 774 


.97 946 


.49 828 


.50 172 


.02 054 


.52 226 


31 


30 


9.47 814 


9.97 942 


9.49 872 


0.50 128 


0.02 058 


0.52 186 


30 


31 


.47 854 


.97 938 


.49 916 


.50084 


.02 062 


.52 146 


29 


32 


.47 894 


.97 934 


.49 960 


.50040 


.02 066 


.52 106 


28 


33 


.47 934 


.97 930 


.50 004 


.49 996 


.02 070 


.52 066 


27 


34 


.47 974 


.97 926 


.50 048 


.49 952 


.02 074 


.52 026 


26 


35 


9.48 014 


9.97 922 


9.50 092 


0.49 908 


0.02 078 


0.51 986 


25 


30 


.48 054 


.97 918 


.50 136 


.49864 


.02 082 


.51 946 


24 


37 


.48 094 


.97 914 


.50 180 


.49 820 


.02 086 


.51 906 


23 


38 


.48 133 


.97 910 


.50 223 


.49 777 


.02 090 


.51 867 


22 


39 


.48 173 


.97 906 


.50 267 


.49 733 


.02 094 


.51 827 


21 


40 


9.48 213 


9.97 902 


9.50311 


0.49 689 


0.02 098 


0.51 787 


20 


41 


.48 252 


.97 898 


.50 355 


.49 645 


.02 102 


.51 748 


19 


42 


.48 292 


.97 894 


.50 398 


.49 602 


.02 106 


.51 708 


18 


43 


.48 332 


.97 890 


.50442 


.49 558 


.02 110 


.51 668 


17 


44 


.48 371 


.97 886 


.50 485 


.49 515 


.02 114 


.51 629 


16 


45 


9.48411 


9.97 882 


9.50 529 


0.49 471 


0.02 118 


0.51 589 


15 


46 


.48 450 


.97 878 


.50 572 


.49 428 


.02 122 


.51 550 


14 


47 


.48 490 


.97 874 


.50 616 


.49384 


.02 126 


.51 510 


13 


48 


.48 529 


.97 870 


.50 659 


.49 341 


.02 130 


.51 471 


12 


49 


.48 568 


.97 866 


.50 703 


.49 297 


.02 134 


.51 432 


11 


50 


9.48 607 


9.97 861 


9.50 746 


0.49 254 


0.02 139 


0.51 393 


10 


51 


.48 647 


.97 857 


.50 789 


.49211 


.02 143 


.51 353 


9 


52 


.48 686 


.97853 


.50833 


.49 167 


.02 147 


.51 314 


8 


53 


.48 725 


.97849 


.50 876 


.49 124 


.02 151 


.51 275 


7 


54 


.48764 


.97845 


.50 919 


.49 081 


.02 155 


.51 236 


6 


55 


9.48 803 


9.97 841 


9.50 962 


0.49 038 


0.02 159 


0.51 197 


5 


56 


.48842 


.97 837 


.51 005 


.48 995 


.02 163 


.51 158 


4 


57 


.48 881 


.97 833 


.51048 


.48 952 


.02 167 


.51 119 


3 


58 


.48 920 


.97 829 


.51 092 


.48 908 


.02 171 


.51080 


2 


59 


.48 959 


.97 825 


.51 135 


.48 865 


.02 175 


.51041 


1 


60 


9.48 998 


9.97 821 


9.51 178 


0.48 822 


0.02 179 


0.51 002 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



107 (287) 



(252) 72 



214 



Table 4. Trigonometric Logarithms 



18 (198) 



(341) 161 



' 


Sin 


Cos 


Tan 


Cot 


See 


Csc 







9.48 998 


9.97 821 


9.51 178 


0.48 822 


0.02 179 


0.51 002 


60 


1 


.49 037 


.97 817 


.51 221 


.48 779 


.02 183 


.50 963 


59 


2 


.49 076 


.97 812 


.51264 


.48 736 


.02 188 


.50 924 


58 


3 


.49 115 


.97 808 


.51 306 


.48 694 


.02 192 


.50 885 


57 


4 


.49 153 


.97804 


.51 349 


.48651 


.02 196 


.50847 


56 


5 


9.49 192 


9.97 800 


9.51 392 


0.48 608 


0.02 200 


0.50 808 


55 


6 


.49 231 


.97 796 


.51 435 


.48 565 


.02204 


.50 769 


54 


7 


.49 269 


.97 792 


.51 478 


.48 522 


.02 208 


.50 731 


53 


8 


.49 308 


.97 788 


.51 520 


.48 480 


.02 212 


.50 692 


52 


9 


.49 347 


.97784 


.51 563 


.48 437 


.02 216 


.50 653 


51 


10 


9.49 385 


9.97 779 


9.51 606 


0.48 394 


0.02 221 


0.50 615 


50 


11 


.49 424 


.97 775 


.51648 


.48 352 


.02 225 


.50 576 


49 


12 


.49 462 


.97 771 


.51 691 


.48 309 


.02 229 


.50 538 


48 


13 


.49 500 


.97 767 


.51 734 


.48 266 


.02 233 


.50 500 


47 


14 


.49 539 


.97 763 


.51 776 


.48 224 


.02 237 


.50 461 


46 


15 


9.49 577 


9.97 759 


9.51 819 


0.48 181 


0.02 241 


0.50 423 


45 


16 


.49 615 


.97 754 


.51 861 


.48 139 


.02 246 


.50 385 


44 


17 


.49 654 


.97 750 


.51 903 


.48 097 


.02 250 


.50 346 


43 


18 


.49 692 


.97 746 


.51 946 


.48 054 


.02 254 


.50 308 


42 


19 


.49 730 


.97 742 


.51 988 


.48 012 


.02 258 


.50 270 


41 


20 


9.49 768 


9.97 738 


9.52 031 


0.47 969 


0.02 262 


0.50 232 


40 


21 


.49 806 


.97 734 


.52 073 


.47 927 


.02 266 


.50 194 


39 


22 


.49844 


.97 729 


.52 115 


.47885 


.02 271 


.50 156 


38 


23 


.49 882 


.97 725 


.52 157 


.47843 


.02 275 


.50 118 


37 


24 


.49 920 


.97 721 


.52200 


.47800 


.02 279 


.50 080 


36 


25 


9.49 958 


9.97 717 


9.52 242 


0.47 758 


0.02 283 


0.50 042 


35 


26 


.49 996 


.97 713 


.52284 


.47 716 


.02 287 


.50 004 


34 


27 


.50 034 


.97 708 


.52 326 


.47 674 


.02 292 


.49 966 


33 


28 


.50 072 


.97704 


.52 368 


.47 632 


.02 296 


.49 928. 


32 


29 


.50 110 


.97 700 


.52 410 


.47 590 


.02 300 


.49 890 


31 


30 


9.50 148 


9.97 696 


9.52 452 


0.47 548 


0.02 304 


0.49 852 


30 


31 


.50 185 


.97 691 


.52 494 


.47 506 


.02 309 


.49 815 


29 


32 


.50 223 


.97 687 


.52 536 


.47464 


.02 313 


.49 777 


28 


33 


.50 261 


.97 683 


.52 578 


.47 422 


.02 317 


.49 739 


27 


34 


.50 298 


.97 679 


.52 620 


.47 380 


.02 321 


.49 702 


26 


35 


9.50 336 


9.97 674 


9.52 661 


0.47 339 


0.02 326 


0.49 664 


25 


36 


.50 374 


.97 670 


.52 703 


.47 297 


.02 330 


.49 626 


24 


37 


.50411 


.97 666 


.52 745 


.47 255 


.02 334 


.49 589 


23 


38 


.50 449 


.97 662 


.52 787 


.47 213 


.02 338 


.49 551 


22 


39 


.50 486 


.97 657 


.52 829 


.47 171 


.02 343 


.49 514 


21 


40 


9.50 523 


9.97 653 


9.52 870 


0.47 130 


0.02 347 


0.49 477 


20 


41 


.50 561 


.97 649 


.52 912 


.47 088 


.02 351 


.49 439 


19 


42 


.50 598 


.97645 


.52 953 


.47047 


.02 355 


.49 402 


18 


43 


.50 635 


.97640 


.52 995 


.47 005 


.02 360 


.49 365 


17 


44 


.50 673 


.97 636 


.53 037 


.46 963 


.02364 


.49 327 


16 


45 


9.50 710 


9.97 632 


9.53 078 


0.46 922 


0.02 368 


0.49 290 


15 


46 


.50 747 


.97 628 


.53 120 


.46 880 


.02 372 


.49 253 


14 


47 


.50784 


.97 623 


.53 161 


.46839 


.02 377 


.49 216 


13 


48 


.50 821 


.97 619 


.53 202 


.46 798 


.02 381 


.49 179 


12 


49 


.50 858 


.97 615 


.53 244 


.46 756 


.02 385 


.49 142 


11 


50 


9.50 896 


9.97 610 


9.53 285 


0.46 715 


0.02 390 


0.49 104 


10 


51 


.50 933 


.97 606 


.53 327 


.46 673 


.02 394 


.49 067 


9 


52 


.50 970 


.97 602 


.53 368 


.46 632 


.02 398 


.49 030 


8 


53 


.51 007 


.97 597 


.53 409 


.46 591 


.02 403 


.48 993 


7 


54 


.51 043 


.97 593 


.53 450 


.46 550 


.02 407 


.48 957 


6 


55 


9.51 080 


9.97 589 


9.53 492 


0.46 508 


0.02 411 


0.48 920 


5 


56 


.51 117 


.97584 


.53 533 


.46 467 


.02 416 


.48 883 


4 


57 


.51 154 


.97 580 


.53 574 


.46 426 


.02 420 


.48846 


3 


58 


.51 191 


.97 576 


.53 615 


.46 385 


.02 424 


.48 809 


2 


59 


.51 227 


.97 571 


.53 656 


.46344 


.02 429 


.48 773 


1 


60 


9.51 264 


9.97 567 


9.53 697 


0.46 303 


0.02 433 


0.48 736 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



108 (288) 



(251) 71 



Table 4. Trigonometric Logarithms 



215 



19 (199) 



(340) 160 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







9.51 264 


9.97 567 


9.53 697 


0.46 303 


0.02 433 


0.48 736 


60 


1 


.51 301 


.97 563 


.53 738 


.46 262 


.02 437 


.48 699 


59 


2 


.51 338 


.97 558 


.53 779 


.46 221 


.02442 


.48 662 


58 


3 


.51 374 


.97 554 


.53 820 


.46 180 


.02 446 


.48 626 


57 


4 


.51 411 


.97 550 


.53 861 


.46 139 


.02 450 


.48 589 


56 


5 


9.51 447 


9.97 545 


9.53 902 


0.46 098 


0.02 455 


0.48 553 


55 


6 


.51484 


.97 541 


.53 943 


.46 057 


.02 459 


.48 516 


54 


7 


.51 520 


.97 536 


.53984 


.46 016 


.02 464 


.48 480 


53 


8 


.51 557 


.97 532 


.54 025 


.45 975 


.02 468 


.48443 


52 


9 


.51 593 


.97 528 


.54065 


.45935 


.02 472 


.48 407 


51 


10 


9.51 629 


9.97 523 


9.54 106 


0.45 894 


0.02 477 


0.48 371 


50 


11 


.51 666 


.97 519 


.54147 


.45853 


.02 481 


.48 334 


49 


12 


.51 702 


.97 515 


.54 187 


.45 813 


.02 485 


.48 298 


48 


13 


.51 738 


.97 510 


.54 228 


.45 772 


.02 490 


.48 262 


47 


14 


.51 774 


.97 506 


.54 269 


.45 731 


.02 494 


.48 226 


46 


15 


9.51 811 


9.97 501 


9.54 309 


0.45 691 


0.02 499 


0.48 189 


45 


16 


.51 847 


.97 497 


.54350 


.45 650 


.02 503 


.48 153 


44 


17 


.51883 


.97 492 


.54390 


.45 610 


.02 508 


.48 117 


43 


18 


.51 919 


.97488 


.54431 


.45 569 


.02 512 


.48 081 


42 


19 


.51 955 


.97484 


.54 471 


.45 529 


.02 516 


.48 045 


41 


20 


9.51 991 


9.97 479 


9.54 512 


0.45 488 


0.02 521 


0.48 009 


40 


21 


.52 027 


.97 475 


.54 552 


.45448 


.02 525 


.47 973 


39 


22 


.52 063 


.97 470 


.54593 


.45 407 


.02 530 


.47 937 


38 


23 


.52 099 


.97 466 


.54 633 


.45 367 


.02 534 


.47 901 


37 


24 


.52 135 


.97 461 


.54673 


.45 327 


.02 539 


.47 865 


36 


25 


9.52 171 


9.97457 


9.54 714 


0.45 286 


0.02 543 


0.47 829 


35 


26 


.52 207 


.97 453 


.54754 


.45246. 


.02 547 


.47 793 


34 


27 


.52 242 


.97448 


.54 794 


.45206 


.02 552 


.47 758 


33 


28 


.52 278 


.97444 


.54835 


.45 165 


.02 556 


.47 722 


32 


29 


.52 314 


.97 439 


.54 875 


.45 125 


.02 561 


.47 686 


31 


30 


9.52 350 


9.97 435 


9.54 915 


0.45 085 


0.02 565 


0.47 650 


30 


31 


.52 385 


.97 430 


.54 955 


.45045 


.02 570 


.47 615 


29 


32 


.52421 


.97 426 


.54995 


.45 005 


.02 574 


.47 579 


28 


33 


.52 456 


.97421 


.55 035 


.44965 


.02 579 


.47 544 


27 


34 


.52 492 


.97 417 


.55 075 


.44 925 


.02 583 


.47 508 


26 


35 


9.52 527 


9.97 412 


9.55 115 


0.44 885 


0.02 588 


0.47 473 


25 


36 


.52 563 


.97 408 


.55 155 


.44845 


.02 592 


.47 437 


24 


37 


.52 598 


.97 403 


.55 195 


.44805 


.02 597 


.47 402 


23 


38 


.52 634 


.97 399 


.55 235 


.44765 


.02 601 


.47 366 


22 


39 


.52 669 


.97 394 


.55 275 


.44725 


.02 606 


.47 331 


21 


40 


9.52 705 


9.97 390 


9.55 315 


0.44685 


0.02 610 


0.47 295 


20 


41 


.52 740 


.97 385 


.55 355 


.44645 


.02 615 


.47 260 


19 


42 


.52 775 


.97 381 


.55 395 


.44605 


.02 619 


.47 225 


18 


43 


.52811 


.97 376 


.55 434 


.44566 


.02 624 


.47 189 


17 


44 


.52846 


.97 372 


.55 474 


.44526 


.02 628 


.47 154 


16 


45 


9.52 881 


9.97 367 


9.55 514 


0.44 486 


0.02 633 


0.47 119 


15 


46 


.52 916 


.97 363 


.55554 


.44 446 


.02 637 


.47084 


14 


47 


.52 951 


.97 358 


.55 593 


.44407 


.02642 


.47049 


13 


48 


.52 986 


.97 353 


.55 633 


.44367 


.02647 


.47 014 


12 


49 


.53 021 


.97 349 


.55 673 


.44327 


.02 651 


.46 979 


11 


50 


9.53 056 


9.97 344 


9.55 712 


0.44 288 


0.02 656 


0.46 944 


10 


51 


.53 092 


.97 340 


.55 752 


.44248 


.02 660 


.46 908 


9 


52 


.53 126 


.97 335 


.55 791 


.44209 


.02 665 


.46 874 


8 


53 


.53 161 


.97 331 


.55831 


.44169 


.02 669 


.46 839 


7 


54 


.53 196 


.97 326 


.55 870 


.44 130 


.02 674 


.46804 


6 


55 


9.53 231 


9.97 322 


9.55 910 


0.44 090 


0.02 678 


0.46 769 


5 


56 


.53 266 


.97 317 


.55 949 


.44051 


.02 683 


.46 734 


4 


57 


.53 301 


.97 312 


.55 989 


.44011 


.02688 


.46 699 


3 


58 


.53 336 


.97 308 


.56 028 


.43 972 


.02 692 


.46664 


2 


59 


.53 370 


.97 303 


.56 067 


.43 933 


.02 697 


.46 630 


1 


60 


9.53 405 


9.97 299 


9.56 107 


9.43 893 


0.02 701 


0.46 595 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



109 (289) 



(250) 70 



216 



Table 4. Trigonometric Logarithms 



20 (200) 



(339) 159 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







9.53 405 


9.97 299 


9.56 107 


0.43 893 


0.02 701 


0.46 595 


60 


1 


.53 440 


.97 294 


.56 146 


.43 854 


.02 706 


.46 560 


59 


2 


.53 475 


.97 289 


.56 185 


.43 815 


.02711 


.46 525 


58 


3 


.53 509 


.97 285 


.56 224 


.43 776 


.02 715 


.46 491 


57 


4 


.53 544 


.97 280 


.56 264 


.43 736 


.02 720 


.46 456 


56 


5 


9.53 578 


9.97 276 


9.56 303 


0.43 697 


0.02 724 


0.46 422 


55 


6 


.53 613 


.97 271 


.56 342 


.43 658 


.02 729 


.46 387 


54 


7 


.53 647 


.97 266 


.56 381 


.43 619 


.02 734 


.46 353 


53 


8 


.53 682 


.97 262 


.56 420 


.43 580 


.02 738 


.46 318 


52 


9 


.53 716 


.97 257 


.56 459 


.43 541 


.02 743 


.46284 


51 


10 


9.53 751 


9.97 252 


9.56 498 


0.43 502 


0.02 748 


0.46 249 


50 


11 


.53 785 


.97 248 


.56 537 


.43 463 


.02 752 


.46 215 


49 


12 


.53 819 


.97 243 


.56 576 


.43 424 


.02 757 


.46 181 


48 


13 


.53 854 


.97 238 


.56 615 


.43 385 


.02 762 


.46 146 


47 


14 


.53 888 


.97 234 


.56 654 


.43 346 


.02 766 


.46 112 


46 


15 


9.53 922 


9.97 229 


9.56 693 


0.43 307 


0.02 771 


0.46 078 


45 


16 


.53 957 


.97 224 


.56 732 


.43 268 


.02 776 


.46 043 


44 


17 


.53 991 


.97 220 


.56 771 


.43 229 


.02 780 


.46 009 


43 


18 


.54 025 


.97 215 


.56 810 


.43 190 


.02 785 


.45 975 


42 


19 


.54 059 


.97 210 


.56849 


.43 151 


.02 790 


.45 941 


41 


20 


9.54 093 


9.97 206 


9.56 887 


0.43 113 


0.02 794 


0.45 907 


40 


21 


.54 127 


.97 201 


.56 926 


.43 074 


.02 799 


.45 873 


39 


22 


.54 161 


.97 196 


.56 965 


.43 035 


.02 804 


.45 839 


38 


23 


.54 195 


.97 192 


.57 004 


.42 996 


.02 808 


.45 805 


37 


24 


.54 229 


.97 187 


.57 042 


.42 958 


.02 813 


.45 771 


36 


25 


9.54 263 


9.97 182 


9.57 081 


0.42 919 


0.02 818 


0.45 737 


35 


26 


.54 297 


.97 178 . 


.57 120 


.42 880 


.02 822 


.45 703 


34 


27 


.54 331 


.97 173 


.57 158 


.42842 


.02 827 


.45 669 


33 


28 


.54 365 


.97 168 


.57 197 


.42 803 


.02 832 


.45 635 


32 


29 


.54 399 


.97 163 


.57 235 


.42 765 


.02 837 


.45 601 


31 


30 


9.54 433 


9.97 159 


9.57 274 


0.42 726 


0.02 841 


0.45 567 


30 


31 


.54466 


.97 154 


.57 312 


.42 688 


.02846 


.45 534 


29 


32 


.54 500 


.97 149 


.57 351 


.42649 


.02 851 


.45 500 


28 


33 


.54 534 


.97 145 


.57 389 


.42611 


.02 855 


.45 466 


27 


34 


.54 567 


.97 140 


.57 428 


.42 572 


.02 860 


.45 433 


26 


35 


9.54 601 


9.97 135 


9.57 466 


0.42 534 


0.02 865 


0.45 399 


25 


36 


.54 635 


.97 130 


.57504 


.42 496 


.02 870 


.45 365 


24 


37 


.54 668 


.97 126 


.57 543 


.42 457 


.02 874 


.45 332 


23 


38 


.54 702 


.97 121 


.57 581 


.42 419 


.02 879 


.45 298 


22 


39 


.54 735 


.97 116 


.57 619 


.42 381 


.02 884 


.45 265 


21 


40 


9.54 769 


9.97 111 


9.57 658 


0.42 342 


0.02 889 


0.45 231 


20 


41 


.54 802 


.97 107 


.57 696 


.42 304 


.02 893 


.45 198 


19 . 


42 


.54 836 


.97 102 


.57 734 


.42 266 


.02 898 


.45 164 


18 


43 


.54869 


.97 097 


.57 772 


.42 228 


.02 903 


.45 131 


17 


44 


.54 903 


.97 092 


.57 810 


.42 190 


.02 908 


.45 097 


16 


45 


9.54 936 


9.97 087 


9.57 849 


0.42 151 


0.02 913 


0.45 064 


15 


46 


.54 969 


.97 083 


.57 887 


.42 113 


.02 917 


.45 031 


14 


47 


.55 003 


.97 078 


.57 925 


.42 075 


.02 922 


.44 997 


13 


48 


.55 036 


.97 073 


.57 963 


.42 037 


.02 927 


.44 964 


12 


49 


.55 069 


.97 068 


.58 001 


.41 999 


.02 932 


.44 931 


11 


50 


9.55 102 


9.97 063 


9.58 039 


0.41 961 


0.02 937 


0.44 898 


10 


51 


.55 136 


.97 059 


.58 077 


.41 923 


.02 941 


.44864 


9 


52 


.55 169 


.97 054 


.58 115 


.41 885 


.02 946 


.44 831 


8 


53 


.55 202 


.97 049 


.58 153 


.41 847 


.02 951 


.44 798 


7 


54 


.55 235 


.97 044 


.58 191 


.41 809 


.02 956 


.44 765 


6 


55 


9.55 268 


9.97 039 


9.58 229 


0.41 771 


0.02 961 


0.44 732 


5 


56 


.55 301 


.97 035 


.58 267 


.41 733 


.02 965 


.44 699 


4 


57 


.55 334 


.97 030 


.58 304 


.41 696 


.02 970 


.44 666 


3 


58 


.55 367 


.97 025 


.58 342 


.41 658 


.02 975 


.44633 


2 


59 


.55 400 


.97 020 


.58 380 


.41 620 


.02 980 


.44 600 


1 


60 


9.55 433 


9.97 015 


9.58418 


0.41 582 


0.02 985 


0.44 567 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



110 (290) 



(249) 69 



Table 4. Trigonometric Logarithms 



217 



21 (201) 



(338) 158 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







9.55 433 


9.97 015 


9.58418 


0.41 582 


0.02 985 


0.44 567 


60 


1 


.55 466 


.97 010 


.58 455 


.41 545 


.02990 


.44534 


59 


2 


.55 499 


.97 005 


.58 493 


.41 507 


.02 995 


.44501 


58 


3 


.55 532 


.97 001 


.58 531 


.41 469 


.02 999 


.44468 


57 


4 


.55 564 


.96 996 


.58 569 


.41 431 


.03 004 


.44436 


56 


5 


9.55 597 


9.96 991 


9.58 606 


0.41 394 


0.03 009 


0.44 403 


55 


6 


.55 630 


.96 986 


.58 644 


.41 356 


.03 014 


.44370 


54 


7 


.55 663 


.96 981 


.58 681 


.41 319 


.03 019 


.44 337 


53 


8 


.55 695 


.96 976 


.58 719 


.41 281 


.03 024 


.44 305 


52 


9 


.55 728 


.96 971 


.58 757 


.41 243 


.03 029 


.44272 


51 


10 


9.55 761 


9.96 966 


9.58 794 


0.41 206 


0.03 034 


0.44 239 


50 


11 


.55 793 


.96 962 


.58 832 


.41 168 


.03 038 


.44207 


49 


12 


.55 826 


.96 957 


.58 869 


.41 131 


.03 043 


.44 174 


48 


13 


.55 858 


.96 952 


.58 907 


.41 093 


.03 048 


.44 142 


47 


14 


.55 891 


.96 947 


.58944 


.41 056 


.03 053 


.44109 


46 


15 


9.55 923 


9.96 942 


9.58 981 


0.41 019 


0.03 058 


0.44 077 


45 


16 


.55 956 


.96 937 


.59 019 


.40 981 


.03 063 


.44044 


44 


17 


.55 988 


.96 932 


.59 056 


.40 944 


.03 068 


.44 012 


43 


18 


.56 021 


.96 927 


.59 094 


.40 906 


.03 073 


.43 979 


42 


19 


.56 053 


.96 922 


.59 131 


.40 869 


.03 078 


' .43947 


41 


20 


9.56 085 


9.96 917 


9.59 168 


0.40 832 


0.03 083 


0.43 915 


40 


21 


.56 118 


.96 912 


.59 205 


.40 795 


.03 088 


.43 882 


39 


22 


.56 150 


.96 907 


.59 243 


.40 757 


.03 093 


.43 850 


38 


23 


.56 182 


.96 903 


.59 280 


.40 720 


.03 097 


.43 818 


37 


24 


.56 215 


.96 898 


.59 317 


.40 683 


.03 102 


.43 785 


36 


25 


9.56 247 


9.96 893 


9.59 354 


0.40 646 


0.03 107 


0.43 753 


35 


26 


.56 279 


.96 888 


.59 391 


.40 609 


.03 112 


.43 721 


34 


27 


.56311 


.96 883 


.59 429 


.40 571 


.03 117 


.43 689 


33 


28 


.56 343 


.96 878 


.59 466 


.40 534 


.03 122 


.43 657 


32 


29 


.56 375 


.96 873 


.59 503 


.40 497 


.03 127 


.43 625 


31 


30 


9.56 408 


9.96 868 


9.59 540 


0.40 460 


0.03 132 


0.43 592 


30 


31 


.56440 


.96 863 


.59 577 


.40 423 


.03 137 


.43 560 


29 


32 


.56 472 


.96 858 


.59 614 


.40 386 


.03 142 


.43 528 


28 


33 


.56 504 


.96853 


.59 651 


.40 349 


.03 147 


.43 496 


27 


34 


.56 536 


.96848 


.59 688 


.40 312 


.03 152 


.43 464 


26 


35 


9.56 568 


9.96 843 


9.59 725 


0.40 275 


0.03 157 


0.43 432 


25 


36 


.56 599 


.96 838 


.59 762 


.40 238 


.03 162 


.43 401 


24 


37 


.56 631 


.96833 


.59 799 


.40 201 


.03 167 


.43 369 


23 


38 


.56 663 


.96 828 


.59 835 


.40 165 


.03 172 


.43 337 


22 


39 


.56 695 


.96 823 


.59 872 


.40 128 


.03 177 


.43 305 


21 


40 


9.56 727 


9.96 818 


9.59 909 


0.40 091 


0.03 182 


0.43 273 


20 


41 


.56 759 


.96 813 


.59 946 


.40 054 


.03 187 


.43 241 


19 


42 


.56 790 


.96 808 


.59 983 


.40 017 


.03 192 


.43 210 


18 


43 


.56 822 


.96 803 


.60 019 


.39 981 


.03 197 


.43 178 


17 


44 


.56 854 


.96 798 


.60 056 


.39944 


.03 202 


.43 146 


16 


45 


9.56 886 


9.96 793 


9.60 093 


0.39 907 


0.03 207 


0.43 114 


15 


46 


.56 917 


.96 788 


.60 130 


.39 870 


.03 212 


.43 083 


14 


47 


.56 949 


.96 783 


.60 166 


.39 834 


.03 217 


.43 051 


13 


48 


.56 980 


.96 778 


.60 203 


.39 797 


.03 222 


.43 020 


12 


49 


.57 012 


.96 772 


.60 240 


.39 760 


.03 228 


.42 988 


11 


50 


9.57 044 


9.96 767 


9.60 276 


0.39 724 


0.03 233 


0.42 956 


10 


51 


.57 075 


.96 762 


.60 313 


.39 687 


.03 238 


.42 925 


9 


52 


.57 107 


.96 757 


.60 349 


.39 651 


.03 243 


.42 893 


8 


53 


.57 138 


.96 752 


.60 386 


.39 614 


.03 248 


.42 862 


7 


54 


.57 169 


.96 747 


.60 422 


.39 578 


.03 253 


.42831 


6 


55 


9.57 201 


9.96 742 


9.60 459 


0.39 541 


0.03 258 


0.42 799 


5 


56 


.57 232 


.96 737 


.60 495 


.39 505 


.03 263 


.42 768 


4 


57 


.57 264 


.96 732 


.60 532 


.39 468 


.03 268 


.42 736 


3 


58 


.57 295 


.96 727 


.60 568 


.39 432 


.03 273 


.42 705 


2 


59 


.57 326 


.96 722 


.60 605 


.39 395 


.03 278 


.42 674 


1 


60 


9.57 358 


9.96 717 


9.60 641 


0.39 359 


0.03 283 


0.42 642 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



111 (291) 



(248) 68 



218 



Table 4. Trigonometric Logarithms 



22 (202) 



(337) 157 



/ 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







9.57 358 


9.96 717 


9.60 641 


0.39 359 


0.03 283 


0.42 642 


60 


1 


.57 389 


.96 711 


.60 677 


.39 323 


.03 289 


.42611 


59 


2 


.57 420 


.96 706 


.60 714 


.39 286 


.03 294 


.42 580 


58 


3 


.57 451 


.96 701 


.60 750 


.39 250 


.03 299 


.42 549 


57 


4 


.57 482 


.96 696 


.60 786 


.39 214 


.03304 


.42 518 


56 


5 


9.57 514 


9.96 691 


9.60 823 


0.39 177 


0.03 309 


0.42 486 


55 


6 


.57 545 


.96 686 


.60859 


.39 141 


.03 314 


.42 455 


54 


7 


.57 576 


.96 681 


.60 895 


.39 105 


.03 319 


.42 424 


53 


8 


.57 607 


.96 676 


.60 931 


.39 069 


.03 324 


.42 393 


52 


9 


.57 638 


.96 670 


.60 967 


.39 033 


.03 330 


.42 362 


51 


10 


9.57 669 


9.96 665 


9.61 004 


9.38 996 


0.03 335 


0.42 331 


50 


11 


.57 700 


.96 660 


.61040 


.38 960 


.03 340 


.42 300 


49 


12 


.57 731 


.96 655 


.61 076 


.38 924 


.03 345 


.42 269 


48 


13 


.57 762 


.96 650 


.61 112 


.38 888 


.03 350 


.42 238 


47 


14 


.57 793 


.96645 


.61 148 


.38 852 


.03 355 


.42 207 


46 


15 


9.57 824 


9.96 640 


9.61 184 


0.38 816 


0.03 360 


0.42 176 


45 


16 


.57 855 


.96 634 


.61 220 


.38 780 


.03 366 


.42 145 


44 


17 


.57 885 


.96 629 


.61 256 


.38744 


.03 371 


.42 115 


43 


18 


.57 916 


.96 624 


.61 292 


.38 708 


.03 376 


.42084 


42 


19 


.57 947 


.96 619 


.61 328 


.38 672 


.03 381 


.42 053 


41 


20 


9.57 978 


9.96 614 


9.61 364 


0.38 636 


0.03 386 


0.42 022 


40 


21 


.58 008 


.96 608 


.61 400 


.38 600 


.03 392 


.41 992 


39 


22 


.58 039 


.96 603 


.61 436 


.38564 


.03 397 


.41 961 


38 


23 


.58 070 


.96 598 


.61 472 


.38 528 


.03 402 


.41 930 


37 


24 


.58 101 


.96 593 


.61 508 


.38 492 


.03 407 


.41 899 


36 


25 


9.58 131 


9.96 588 


9.61 544 


0.38 456 


0.03 412 


0.41 869 


35 


26 


.58 162 


.96 582 


.61 579 


.38 421 


.03 418 


.41 838 


34 


27 


.58 192 


.96 577 


.61 615 


.38 385 


.03 423 


.41 808 


33 


28 


.58 223 


.96 572 


.61 651 


.38 349 


.03 428 


.41 777 


32 


29 


.58 253 


.96 567 


.61 687 


.38313 


.03 433 


.41 747 


31 


30 


9.58 284 


9.96 562 


9.61 722 


0.38 278 


0.03 438 


0.41 716 


30 


31 


.58 314 


.96 556 


.61 758 


.38 242 


.03444 


.41 686 


29 


32 


.58 345 


.96 551 


.61 794 


.38 206 


.03449 


.41 655 


28 


33 


.58 375 


.96 546 


.61830 


.38 170 


.03454 


.41 625 


27 


34 


.58 406 


.96 541 


.61 865 


.38 135 


.03 459 


.41 594 


26 


35 


9.58 436 


9.96 535 


9.61 901 


0.38 099 


0.03 465 


0.41 564 


25 


36 


.58 467 


.96 530 


.61 936 


.38064 


.03 470 


.41 533 


24 


37 


.58 497 


.96 525 


.61 972 


.38 028 


.03 475 


.41 503 


23 


38 


.58 527 


.96 520 


.62 008 


.37 992 


.03 480 


.41 473 


22 


39 


.58 557 


.96 514 


.62043 


.37 957 


.03 486 


.41 443 


21 


40 


9.58 588 


9.96 509 


9.62 079 


0.37 921 


0.03 491 


0.41 412 


20 


41 


.58 618 


.96 504 


.62114 


.37 886 


.03 496 


.41 382 


19 


42 


.58648 


.96 498 


.62 150 


.37 850 


.03 502 


.41 352 


18 


43 


.58 678 


.96 493 


.62 185 


.37 815 


.03 507 


.41 322 


17 


44 


.58 709 


.96 488 


.62 221 


.37 779 


.03 512 


.41 291 


16 


45 


9.58 739 


9.96 483 


9.62 256 


0.37 744 


0.03 517 


0.41 261 


15 


46 


.58 769 


.96 477 


.62 292 


.37 708 


.03 523 


.41 231 


14 


47 


.58 799 


.96 472 


.62 327 


.37 673 


.03 528 


Al 201 


13 


48 


.58 829 


.96 467 


.62 362 


.37 638 


.03 533 


.41 171 


12 


49 


.58 859 


.96 461 


.62 398 


.37 602 


.03 539 


.41 141 


11 


50 


9.58 889 


9.96 456 


9.62 433 


0.37 567 


0.03 544 


0.41 111 


10 


51 


.58 919 


.96 451 


.62 468 


.37 532 


.03 549 


.41 081 


9 


52 


.58 949 


.96445 


.62504 


.37 496 


.03 555 


.41 051 


8 


53 


.58 979 


.96440 


.62 539 


.37 461 


.03 560 


.41 021 


7 


54 


.59 009 


.96 435 


.62 574 


.37 426 


.03 565 


.40 991 


6 


55 


9.59 039 


9.96 429 


9.62 609 


0.37 391 


0.03 571 


0.40 961 


5 


56 


.59 069 


.96 424 


.62 645 


.37 355 


.03 576 


.40 931 


4 


57 


.59 098 


.96 419 


.62 680 


.37 320 


.03 581 


.40 902 


3 


58 


.59 128 


.96 413 


.62 715 


.37 285 


.03 587 


.40 872 


2 


59 


.59 158 


.96 408 


.62 750 


.37 250 


.03 592 


.40842 


1 


60 


9.59 188 


9.96 403 


9.62 785 


0.37 215 


0.03 597 


0.40 812 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



112 r (292) 



(247) 67 



Table 4. Trigonometric Logarithms 



219 



83 (203) 



(336) 156 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







9.59 188 


9.96 403 


9.62 785 


0.37 215 


0.03 597 


0.40 812 


60 


1 


.59 218 


.96 397 


.62 820 


.37 180 


.03 603 


.40 782 


59 


2 


.59 247 


.96 392 


.62855 


.37 145 


.03 608 


.40 753 


58 


3 


.59 277 


.96 387 


.62 890 


.37 110 


.03 613 


.40 723 


57 


4 


.59 307 


.96 381 


.62 926 


.37 074 


.03 619 


.40 693 


56 


5 


9.59 336 


9.96 376 


9.62 961 


0.37 039 


0.03 624 


0.40664 


55 


6 


.59 366 


.96 370 


.62 996 


.37004 


.03 630 


.40 634 


54 


7 


.59 396 


.96 365 


.63 031 


.36 969 


.03 635 


.40604 


53 


8 


.59 425 


.96 360 


.63 066 


.36 934 


.03 640 


.40 575 


52 


9 


.59 455 


.96 354 


.63 101 


.36 899 


.03 646 


.40545 


51 


10 


9.59 484 


9.96 349 


9.63 135 


0.36 865 


0.03 651 


0.40 516 


50 


11 


.59 514 


.96 343 


.63 170 


.36830 


.03 657 


.40 486 


49 


12 


' .59 543 


.96 338 


.63 205 


.36 795 


.03 662 


.40 457 


48 


13 


.59 573 


.96 333 


.63 240 


.36 760 


.03 667 


.40 427 


47 


14 


.59 602 


.96 327 


.63 275 


.36 725 


.03 673 


.40 398 


46 


15 


9.59 632 


9.96 322 


9.63 310 


0.36 690 


0.03 678 


0.40 368 


45 


16 


.59 661 


.96 316 


.63 345 


.36 655 


.03684 


.40 339 


44 


17 


.59 690 


.96311 


.63 379 


.36 621 


.03 689 


.40 310 


43 


18 


.59 720 


.96 305 


.63 414 


.36 586 


.03 695 


.40 280 


42 


19 


.59 749 


.96 300 


.63 449 


.36 551 


.03 700 


.40 251 


41 


20 


9.59 778 


9.96 294 


9.63 484 


0.36 516 


0.03 706 


0.40 222 


40 


21 


.59 808 


.96 289 


.63 519 


.36 481 


.03711 


.40 192 


39 


22 


.59 837 


.96284 


.63553 


.36 447 


.03 716 


.40 163 


38 


23 


.59 866 


.96 278 


.63 588 


.36412 


.03 722 


.40 134 


37 


24 


.59 895 


.96 273 


.63 623 


.36 377 


.03 727 


.40 105 


36 


25 


9.59 924 


9.96 267 


9.63 657 


0.36 343 


0.03 733 


0.40 076 


35 


26 


.59 954 


.96 262 


.63 692 


.36 308 


.03 738 


.40 046 


34 


27 


.59983 


.96 256 


.63 726 


.36 274 


.03 744 


.40 017 


33 


28 


.60 012 


.96 251 


.63 761 


.36 239 


.03 749 


.39 988 


32 


29 


.60 041 


.96 245 


.63 796 


.36 204 


.03 755 


.39 959 


31 


30 


9.60 070 


9.96 240 


9.63 830 


0.36 170 


0.03 760 


.39 930 


30 


31 


.60 099 


.96 234 


.63 865 


.36 135 


.03 766 


.39 901 


29 


32 


.60 128 


.96 229 


.63 899 


.36 101 


.03 771 


.39 872 


28 


33 


.60 157 


.96 223 


.63 934 


.36 066 


.03 777 


.39843 


27 


34 


.60 186 


.96 218 


.63 968 


.36 032 


.03 782 


.39 814 


26 


35 


9.60 215 


9.96 212 


9.64 003 


0.35 997 


0.03 788 


0.39 785 


25 


36 


.60 244 


.96 207 


.64 037 


.35 963 


.03 793 


.39 756 


24 


37 


.60 273 


.96 201 


.64072 


.35 928 


.03 799 


.39 727 


23 


38 


.60 302 


.96 196 


.64 106 


.35 894 


.03804 


.39 698 


22 


39 


.60 331 


.96 190 


.64 140 


.35 860 


.03 810 


.39 669 


21 


40 


9.60 359 


9.96 185 


9.64 175 


0.35 825 


0.03 815 


0.39 641 


20 


41 


.60 388 


.96 179 


.64209 


.35 791 


.03 821 


.39 612 


19 


42 


.60 417 


.96 174 


.64243 


.35 757 


.03 826 


.39 583 


18 


43 


.60446 


.96 168 


.64278 


.35 722 


.03832 


.39 554 


17 


44 


.60 474 


.96 162 


.64 312 


.35688 


.03838 


.39 526 


16 


45 


9.60 503 


9.96 157 


9.64 346 


0.35 654 


0.03 843 


0.39 497 


15 


46 


.60 532 


.96 151 


.64381 


.35 619 


.03849 


.39 468 


14 


47 


.60 561 


.96 146 


.64415 


.35 585 


.03854 


.39 439 


13 


48 


.60589 


.96 140 


.64449 


.35 551 


.03 860 


.39411 


12 


49 


.60 618 


.96 135 


.64483 


.35 517 


.03 865 


.39 382 


11 


50 


9.60 646 


9.96 129 


9.64517 


0.35 483 


0.03 871 


0.39 354 


10 


51 


.60 675 


.96 123 


.64552 


.35448 


.03 877 


.39 325 


9 


52 


.60704 


.96 118 


.64586 


.35 414 


.03 882 


.39 296 


8 


53 


.60 732 


.96112 


.64620 


.35380 


.03888 


.39 268 


7 


54 


.60 761 


.96 107 


.64654 


.35 346 


.03 893 


.39 239 


6 


55 


9.60 789 


9.96 101 


9.64688 


0.35 312 


0.03 899 


0.39211 


5 


56 


.60 818 


.96095 


.64722 


.35 278 


.03 905 


.39 182 


4 


57 


.60846 


.96090 


.64756 


.35244 


.03 910 


.39 154 


3 


58 


.60 875 


.96084 


.64790 


.35 210 


.03 916 


.39 125 


2 


59 


.60 903 


.96 079 


.64824 


.35 176 


.03 921 


.39097 


1 


60 


9.60 931 


9.96 073 


9.64 858 


0.35 142 


0.03 927 


0.39 069 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



113 (293) 



(246) 66 



220 



Table 4. Trigonometric Logarithms 



24 (204) 



(335) 155 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







9.60 931 


9.96 073 


9.64 858 


0.35 142 


0.03 927 


0.39 069 


60 


1 


.60 960 


.96 067 


.64 892 


.35 108 


.03 933 


.39 040 


59 


2 


.60 988 


.96 062 


.64 926 


.35 074 


.03 938 


.39 012 


58 


3 


.61 016 


.96 056 


.64960 


.35 040 


.03 944 


.38 984 


57 


4 


.61045 


.96 050 


.64994 


.35 006 


.03 950 


.38 955 


56 


5 


9.61 073 


9.96 045 


9.65 028 


0.34 972 


0.03 955 


0.38 927 


55 


6 


.61 101 


.96 039 


.65 062 


.34 938 


.03 961 


.38 899 


54 


7 


.61 129 


.96 034 


.65 096 


.34 904 


.03 966 


.38 871 


53 


8 


.61 158 


.96 028 


.65 130 


.34 870 


.03 972 


.38 842 


52 


9 


.61 186 


.96 022 


.65 164 


.34 836 


.03 978 


.38 814 


51 


10 


9.61 214 


9.96 017 


9.65 197 


0.34 803 


0.03 983 


0.38 786 


50 


11 


.61 242 


.96011 


.65 231 


.34 769 


.03 989 


.38 758 


49 


12 


.61 270 


.96 005 


.65 265 


.34 735 


.03 995 


.38 730 


48 


13 


.61 298 


.96 000 


.65 299 


.34 701 


.04000 


.38 702 


47 


14 


.61 326 


.95 994 


.65 333 


.34 667 


.04006 


.38 674 


46 


15 


9.61 354 


9.95 988 


9.65 366 


0.34 634 


0.04 012 


0.38 646 


45 


16 


.61 382 


.95 982 


.65 400 


.34600 


.04018 


.38 618 


44 


17 


.61411 


.95 977 


.65 434 


.34 566 


.04 023 


.38 589 


43 


18 


.61 438 


.95 971 


.65 467 


.34 533 


.04029 


.38 562 


42 


19 


.61 466 


.95 965 


.65 501 


.34 499 


.04035 


.38 534 


41 


20 


9.61 494 


9.95 960 


9.65 535 


0.34 465 


0.04 040 


0.38 506 


40 


21 


.61 522 


.95 954 


.65 568 


.34 432 


.04 046 


.38 478 


39 


22 


.61 550 


.95 948 


.65 602 


.34398 


.04052 


.38 450 


38 


23 


.61 578 


.95 942 


.65 636 


.34 364 


.04058 


.38 422 


37 


24 


.61 606 


.95 937 


.65 669 


.34 331 


.04 063 


.38 394 


36 


25 


9.61 634 


9.95 931 


9.65 703 


0.34 297 


0.04 069 


0.38 366 


35 


26 


.61 662 


.95 925 


.65 736 


.34 264 


.04075 


.38 338 


34 


27 


.61 689 


.95 920 


.65 770 


.34 230 


.04 080 


.38311 


33 


28 


.61 717 


.95 914 


.65 803 


.34 197 


.04 086 


.38 283 


32 


29 


.61 745 


.95 908 


.65 837 


.34163 


.04 092 


.38 255 


31 


30 


9.61 773 


9.95 902 


9.65 870 


0.34 130 


0.04 098 


0.38 227 


30 


31 


.61 800 


.95 897 


.65 904 


.34 096 


.04 103 


.38 200 


29 


32 


.61 828 


.95 891 


.65 937 


.34 063 


.04 109 


.38 172 


28 


33 


.61 856 


.95 885 


.65 971 


.34 029 


.04115 


.38 144 


27 


34 


.61 883 


.95 879 


.66 004 


.33 996 


.04 121 


.38 117 


26 


35 


9.61 911 


9.95 873 


9.66 038 


0.33 962 


0.04 127 


0.38 089 


25 


36 


.61 939 


.95 868 


.66 071 


.33 929 


.04 132 


.38 061 


24 


37 


.61 966 


.95 862 


.66 104 


.33 896 


.04 138 


.38 034 


23 


38 


.61 994 


.95 856 


.66 138 


.33 862 


.04 144 


.38 006 


22 


39 


.62 021 


.95850 


.66 171 


.33 829 


.04 150 


.37 979 


21 


40 


9.62 049 


9.95 844 


9.66 204 


0.33 796 


0.04 156 


0.37 951 


20 


41 


.62 076 


.95 839 


.66 238 


.33 762 


.04 161 


.37 924 


19 


42 


.62 104 


.95833 


.66 271 


.33 729 


.04 167 


.37 896 


18 


43 


.62 131 


.95 827 


.66 304 


.33 696 


.04173 


.37 869 


17 


44 


.62 159 


.95 821 


.66 337 


.33 663 


.04 179 


.37 841 


16 


45 


9.62 186 


9.95 815 


9.66 371 


0.33 629 


0.04 185 


0.37 814 


15 


46 


.62 214 


.95 810 


.66 404 


.33 596 


.04 190 


.37 786 


14 


47 


.62 241 


.95804 


.66 437 


.33 563 


.04 196 


.37 759 


13 


48 


.62 268 


.95 798 


.66 470 


.33 530 


'.04 202 


.37 732 


12 


49 


.62 296 


.95 792 


.66 503 


.33 497 


.04 208 


.37 704 


11 


50 


9.62 323 


9.95 786 


9.66 537 


0.33 463 


0.04 214 


0.37 677 


10 


51 


.62 350 


.95 780 


.66 570 


.33 430 


.04220 


.37 650 


9 


52 


.62 377 


.95 775 


.66 603 


.33 397 


.04225 


.37 623 


8 


53 


.62 405 


.95 769 


.66 636 


.33 364 


.04231 


.37 595 


7 


54 


.62 432 


.95 763 


.66 669 


.33 331 


.04 237 


.37 568 


6 


55 


9.62 459 


9.95 757 


9.66 702 


0.33 298 


0.04 243 


0.37 541 


5 


56 


.62 486 


.95 751 


.66 735 


.33 265 


.04 249 


.37 514 


4 


57 


.62 513 


.95 745 


.66 768 


.33 232 


.04 255 


.37 487 


3 


58 


.62 541 


.95 739 


.66801 


.33 199 


.04261 


.37 459 


2 


59 


.62 568 


.95 733 


.66 834 


.33 166 


.04 267 


.37 432 


1 


60 


9.62 595 


9.95 728 


9.66 867 


0.33 133 


0.04 272 


0.37 405 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



114 (294) 



(245) 65 



Table 4. Trigonometric Logarithms 



221 



25 (205) 



(334) 154 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







9.62 595 


9.95 728 


9.66 867 


0.33 133 


0.04 272 


0.37 405 


60 


1 


.62 622 


.95 722 


.66 900 


.33 100 


.04278 


.37 378 


59 


2 


.62649 


.95 716 


.66 933 


.33 067 


.04284 


.37 351 


58 


3 


.62 676 


.95 710 


.66 966 


.33 034 


.04290 


.37 324 


57 


4 


.62 703 


.95 704 


.66999 


.33 001 


.04296 


.37 297 


56 


5 


9.62 730 


9.95 698 


9.67 032 


0.32 968 


0.04 302 


0.37 270 


55 


6 


.62 757 


.95 692 


.67 065 


.32 935 


.04 308 


.37 243 


54 


7 


.62784 


.95 686 


.67 098 


.32 902 


.04 314 


.37 216 


53 


8 


.62811 


.95 680 


.67 131 


.32 869 


.04 320 


.37 189 


52 


9 


.62838 


.95 674 


.67 163 


.32 837 


.04 326 


.37 162 


51 


10 


9.62 865 


9.95 668 


9.67 196 


0.32 804 


0.04 332 


0.37 135 


50 


11 


.62 892 


.95 663 


.67 229 


.32 771 


.04 337 


.37 108 


49 


12 


.62 918 


.95 657 


.67 262 


.32 738 


.04343 


.37 082 


48 


13 


.62 945 


.95 651 


.67 295 


.32 705 


.04 349 


.37 055 


47 


14 


.62 972 


.95 645 


.67 327 


.32 673 


.04 355 


.37 028 


46 


15 


9.62 999 


9.95 639 


9.67 360 


0.32 640 


0.04 361 


0.37 001 


45 


16 


.63 026 


.95 633 


.67 393 


.32 607 


.04 367 


.36 974 


44 


17 


.63 052 


.95 627 


.67 426 


.32 574 


.04 373 


.36 948 


43 


18 


.63 079 


.95 621 


.67 458 


.32 542 


.04379 


.36 921 


42 


19 


.63 106 


.95 615 


.67 491 


.32 509 


.04 385 


.36 894 


41 


20 


9.63 133 


9.95 609 


9.67 524 


0.32 476 


0.04 391 


0.36 867 


40 


21 


.63 159 


.95 603 


.67 556 


.32 444 


.04 397 


.36 841 


39 


22 


.63 186 


.95 597 


.67 589 


.32411 


.04 403 


.36 814 


38 


23 


.63 213 


.95 591 


.67 622 


.32 378 


.04 409 


.36 787 


37 


24 


.63 239 


.95 585 


.67 654 


.32 346 


.04415 


.36 761 


36 


25 


9.63 266 


9.95 579 


9.67 687 


0.32 313 


0.04 421 


0.36 734 


35 


26 


.63 292 


.95 573 


.67 719 


.32 281 


.04 427 


.36 708 


34 


27 


.63 319 


.95 567 


.67 752 


.32 248 


.04 433 


.36 681 


33 


28 


.63 345 


.95 561 


.67 785 


.32 215 


.04 439 


.36 655 


32 


29 


.63 372 


.95 555 


.67 817 


.32 183 


.04445 


.36 628 


31 


30 


9.63 398 


9.95 549 


9.67 850 


0.32 150 


0.04 451 


0.36 602 


30 


31 


.63 425 


.95 543 


.67 882 


.32 118 


.04 457 


.36 575 


29 


32 


.63 451 


.95 537 


.67 915 


.32 085 


.04 463 


.36 549 


28 


33 


.63 478 


.95 531 


.67 947 


.32 053 


.04 469 


.36 522 


27 


34 


.63 504 


.95 525 


.67 980 


.32 020 


.04 475 


.36 496 


26 


35 


9.63 531 


9.95 519 


9.68 012 


0.31 988 


0.04 481 


0.36 469 


25 


36 


.63 557 


.95 513 


.68 044 


.31 956 


.04 487 


.36 443 


24 


37 


.63583 


.95 507 


.68 077 


.31 923 


.04 493 


.36 417 


23 


38 


.63 610 


.95500 


.68 109 


.31 891 


.04 500 


.36 390 


22 


39 


.63 636 


.95 494 


.68 142 


.31 858 


.04 506 


.36 364 


21 


40 


9.63 662 


9.95 488 


9.68 174 


0.31 826 


0.04 512 


0.36 338 


20 


41 


.63 689 


.95 482 


.68 206 


.31 794 


.04 518 


. 36311 


19 


42 


.63 715 


.95 476 


.68 239 


.31 761 


.04 524 


.36 285 


18 


43 


.63 741 


.95 470 


.68 271 


.31 729 


.04530 


.36 259 


17 


44 


.63 767 


.95 464 


.68 303 


.31 697 


.04536 


.36 233 


16 


45 


9.63 794 


9.95 458 


9.68 336 


0.31 664 


0.04 542 


0.36 206 


15 


46 


.63 820 


.95 452 


.68 368 


.31 632 


.04548 


.36 180 


14 


47 


.63846 


.95 446 


.68 400 


.31 600 


.04 554 


.36 154 


13 


48 


.63 872 


.95 440 


.68 432 


.31 568 


.04 560 


.36 128 


12 


49 


.63 898 


.95 434 


.68 465 


.31 535 


.04 566 


.36 102 


11 


50 


9.63 924 


9.95 427 


9.68 497 


0.31 503 


0.04 573 


0.36 076 


10 


51 


.63 950 


.95 421 


.68 529 


.31 471 


.04 579 


.36 050 


9 


52 


.63 976 


.95 415 


.68 561 


.31 439 


.04585 


.36 024 


8 


53 


.64002 


.95 409 


.68 593 


.31 407 


.04591 


.35 998 


7 


54 


.64028 


.95 403 


.68 626 


.31 374 


.04597 


.35 972 


6 


55 


9.64 054 


9.95 397 


9.68 658 


0.31 342 


0.04 603 


0.35 946 


5 


56 


.64080 


.95 391 


.68 690 


.31 310 


.04609 


.35 920 


4 


57 


.64 106 


.95 384 


.68 722 


.31 278 


.04 616 


.35 894 


3 


58 


.64 132 


.95 378 


' .68754 


.31 246 


.04 622 


.35 868 


2 


59 


.64 158 


.95 372 


.68 786 


.31 214 


.04 628 


.35842 


1. 


60 


9.64 184 


9.95 366 


9.68 818 


0.31 182 


0.04 634 


0.35 816 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



115 (295) 



(244) 64 



222 



Table 4. Trigonometric Logarithms 



26 (206) 



(333) 153 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







9.64 184 


9.95 366 


9.68 818 


0.31 182 


0.04 634 


0.35 816 


60 


1 


.64 210 


.95 360 


.68 850 


.31 150 


.04 640 


.35 790 


59 


2 


.64 236 


.95 354 


.68 882 


.31 118 


.04 646 


.35 764 


58 


3 


.64262 


.95 348 


.68 914 


.31 086 


.04 652 


.35 738 


57 


4 


.64288 


.95 341 


.68 946 


.31054 


.04659 


.35 712 


56 


5 


9.64 313 


9.95 335 


9.68 978 


0.31 022 


0.04 665 


0.35 687 


55 


6 


.64 339 


.95 329 


.69 010 


.30 990 


.04 671 


.35 661 


54 


7 


.64365 


.95 323 


.69 042 


.30 958 


.04677 


.35 635 


53 


8 


.64391 


.95 317 


.69 074 


.30 926 


.04 683 


.35 609 


52 


9 


.64417 


.95 310 


.69 106 


.30 894 


.04690 


.35 583 


51 


10 


9.64 442 


9.95 304 


9.69 138 


0.30 862 


0.04 696 


0.35 558 


50 


11 


.64 468 


.95 298 


.69 170 


.30 830 


.04702 


.35 532 


49 


12 


.64 494 


.95 292 


.69 202 


.30 798 


.04 708 


.35 506 


48 


13 


.64 519 


.95 286 


.69 234 


.30 766 


.04 714 


.35 481 


47 


14 


.64 545 


.95 279 


.69 266 


.30 734 


.04 721 


.35 455 


46 


15 


9.64 571 


9.95 273 


9.69 298 


0.30 702 


0.04 727 


0.35 429 


45 


16 


.64 596 


.95 267 


.69 329 


.30 671 


.04 733 


.35 404 


44 


17 


.64622 


.95 261 


.69 361 


.30 639 


.04739 


.35 378 


43 


18 


.64647 


.95 254 


.69 393 


.30 607 


.04746 


.35 353 


42 


19 


.64673 


.95 248 


.69 425 


.30 575 


.04 752 


.35 327 


41 


20 


9.64 698 


9.95 242 


9.69 457 


0.30 543 


0.04 758 


0.35 302 


40 


21 


.64724 


.95 236 


.69 488 


.30 512 


.04 764 


.35 276 


39 


22 


.64 749 


.95 229 


.69 520 


.30 480 


.04 771 


.35 251 


38 


23 


.64 775 


.95 223 


.69 552 


.30 448 


.04 777 


.35 225 


37 


24 


.64800 


.95 217 


.69584 


.30 416 


.04 783 


.35 200 


36 


25 


9.64 826 


9.95211 


9.69 615 


0.30 385 


0.04 789 


0.35 174 


35 


26 


.64851 


.95 204 


.69 647 


.30 353 


.04 796 


.35 149 


34 


27 


.64877 


.95 198 


.69 679 


.30 321 


.04802 


.35 123 


33 


28 


.64 902 


.95 192 


.69 710 


.30 290 


.04808 


.35 098 


32 


29 


.64927 


.95 185 


.69 742 


.30 258 


.04815 


.35 073 


31 


30 


9.64 953 


9.95 179 


9.69 774 


0.30 226 


0.04 821 


0.35 047 


30 


31 


.64978 


.95 173 


.69 805 


.30 195 


.04 827 


.35 022 


29 


32 


.65 003 


.95 167 


.69 837 


.30 163 


.04 833 


.34 997 


.28 


33 


.65 029 


.95 160 


.69 868 


.30 132 


.04840 


.34 971 


27 


34 


.65 054 


.95 154 


.69 900 


.30 100 


.04846 


.34 946 


26 


35 


9.65 079 


9.95 148 


9.69 932 


0.30 068 


0.04 852 


0.34 921 


25 


36 


.65 104 


.95 141 


.69 963 


.30 037 


.04 859 


.34 896 


24 


37 


.65 130 


.95 135 


.69 995 


.30 005 


.04 865 


.34 870 


23 


38 


.65 155 


.95 129 


.70 026 


.29 974 


.04 871 


.34845 


22 


39 


.65 180 


.95 122 


.70 058 


.29 942 


.04 878 


.34 820 


21 


40 


9.65 205 


9.95 116 


9.70 089 


0.29911 


0.04884 


0.34 795 


20 


41 


.65 230 


.95 110 


.70 121 


.29 879 


.04 890 


.34 770 


19 


42 


.65 255 


.95 103 


.70 152 


.29848 


.04 897 


.34 745 


18 


43 


.65 281 


.95 097 


.70 184 


.29 816 


.04 903 


.34 719 


17 


44 


.65 306 


.95 090 


.70 215 


.29 785 


.04 910 


.34 694 


16 


45 


9.65 331 


9.95 084 


9.70 247 


0.29 753 


0.04 916 


0.34 669 


15 


46 


.65 356 


.95 078 


.70 278 


.29 722 


.04 922 


.34 644 


14 


47 


.65 381 


.95 071 


.70 309 


.29 691 


.04 929 


..4 619 


13 


48 


.65 406 


.95 065 


.70 341 


.29 659 


.04 935 


.34 594 


12 


49 


.65 431 


.95 059 


.70 372 


.29 628 


.04 941 


.34 569 


11 


50 


9.65 456 


9.95 052 


9.70 404 


0.29 596 


0.04 948 


0.34 544 


10 


51 


.65 481 


.95 046 


.70 435 


.29 565 


.04 954 


.34 519 


9 


52 


.65 506 


.95 039 


.70 466 


.29 534 


.04 961 


.34 494 


8 


53 


.65 531 


.95 033 


.70 498 


.29 502 


.04 967 


.34 469 


7 


54 


.65 556 


.95 027 


.70 529 


.29 471 


.04 973 


.34 444 


6 


55 


9.65 580 


9.95 020 


9.70 560 


0.29 440 


0.04 980 


0.34 420 


5 


56 


.65 605 


.95 014 


.70 592 


.29 408 


.04 986 


.34 395 


4 


57 


.65 630 


.95 007 


.70 623 


.29 377 


.04 993 


.34 370 


3 


58 


.65 655 


.95 001 


.70 654 


.29 346 


.04 999 


.34 345 


2 


59 


.65 680 


.94 995 


.70 685 


.29 315 


.05 005 


.34 320 


1 


60 


9.65 705 


9.94 988 


9.70 717 


0.29 283 


0.05 012 


0.34 295 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



116 (296) 



(243) 63 



Table 4. Trigonometric Logarithms 



223 



27 (207) 



(332) 152 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







9.65 705 


9.94 988 


9.70717 


0.29 283 


0.05 012 


0.34 295 


60 


1 


.65 729 


.94 982 


.70 748 


.29 252 


.05 018 


.34 271 


59 


2 


.65 754 


.94 975 


.70 779 


.29 221 


.05 025 


.34 246 


58 


3 


.65 779 


.94 969 


.70 810 


.29 190 


.05 031 


.34 221 


57 


4 . 


.65804 


.94 962 


.70841 


.29 159 


.05 038 


.34 196 


56 


5 


9.65 828 


9.94 956 


9.70 873 


0.29 127 


0.05044 


0.34 172 


55 


6 


.65853 


.94 949 


.70904 


.29 096 


.05 051 


.34 147 


54 


7 


.65 878 


.94943 


.70 935 


.29 065 


.05 057 


.34 122 


53 


8 


.65 902 


.94 936 


.70 966 


.29 034 


.05064 


.34 098 


52 


9 


.65 927 


.94 930 


.70 997 


.29 003 


.05 070 


.34 073 


51 


10 


9.65 952 


9.94 923 


9.71 028 


0.28 972 


0.05 077 


0.34 048 


50 


11 


.65 976 


.94917 


.71 059 


.28 941 


.05 083 


.34 024 


49 


12 


.66001 


.94911 


.71 090 


.28 910 


.05 089 


.33 999 


48 


13 


.66 025 


.94904 


.71 121 


.28 879 


.05 096 


.33 975 


47 


14 


.66 050 


.94 898 


.71 153 


.28847 


.05 102 


.33 950 


46 


15 


9.66 075 


9.94 891 


9.71 184 


0.28 816 


0.05 109 


0.33 925 


45 


16 


.66 099 


.94 885 


.71 215 


.28 785 


.05 115 


.33 901 


44 


17 


.66 124 


.94 878 


.71 246 


.28754 


.05 122 


.33 876 


43 


18 


.66 148 


.94 871 


.71 277 


.28 723 


.05 129 


.33 852 


42 


19 


.66 173 


.94 865 


.71 308 


.28 692 


.05 135 


.33 827 


41 


20 


9.66 197 


9.94 858 


9.71 339 


0.28 661 


0.05 142 


0.33 803 


40 


21 


.66 221 


.94852 


.71 370 


.28 630 


.05 148 


.33 779 


39 


22 


.66 246 


.94845 


.71 401 


.28 599 


.05 155 


.33 754 


38 


23 


.66270 


.94839 


.71 431 


.28 569 


.05 161 


.33 730 


37 


24 


.66 295 


.94832 


.71 462 


.28 538 


.05 168 


.33 705 


36 


25 


9.66 319 


9.94 826 


9.71 493 


0.28 507 


0.05 174 


0.33 681 


35 


26 


.66 343 


.94 819 


.71 524 


.28 476 


.05 181 


.33 657 


34 


27 


.66 368 


.94 813 


.71 555 


.28445 


.05 187 


.33 632 


33 


28 


.66 392 


.94806 


.71 586 


.28.414 


.05 194 


.33 608 


32 


29 


.66416 


.94 799 


.71 617 


.28383 


.05 201 


.33584 


31 


30 


9.66 441 


9.94 793 


9.71 648 


0.28 352 


0.05 207 


0.33 559 


30 


31 


.66 465 


.94 786 


.71 679 


.28 321 


.05 214 


.33 535 


29 


32 


.66 489 


.94 780 


.71 709 


.28 291 


.05 220 


.33511 


28 


33 


.66 513 


.94 773 


.71 740 


.28 260 


.05 227 


.33 487 


27 


34 


.66 537 


.94 767 


.71 771 


.28 229 


.05 233 


.33 463 


26 


35 


9.66 562 


9.94 760 


9.71 802 


0.28 198 


0.05 240 


0.33 438 


25 


36 


.66 586 


.94 753 


.71833 


.28 167 


.05 247 


.33 414 


24 


37 


.66 610 


.94 747 


.71 863 


.28 137 


.05 253 


.33 390 


23 


38 


.66 634 


.94 740 


.71 894 


.28 106 


.05 260 


.33 366 


22 


39 


.66658 


.94 734 


.71 925 


.28 075 


.05 266 


.33 342 


21 


40 


9.66 682 


9.94 727 


9.71 955 


0.28 045 


0.05 273 


0.33 318 


20 


41 


.66 706 


.94 720 


.71 986 


.28 014 


.05 280 


.33 294 


19 


42 


.66 731 


.94 714 


.72 017 


.27 983 


.05 286 


.33 269 


18 


43 


.66 755 


.94 707 


.72048 


.27 952 


.05 293 


.33 245 


17 


44 


.66 779 


.94700 


.72 078 


.27 922 


.05 300 


.33 221 


16 


45 


9.66 803 


9.94 694 


9.72 109 


0.27 891 


0.05 306 


0.33 197 


15 


46 


.66 827 


.94 687 


.72 140 


.27860 


.05 313 


.33 173 


14 


47 


.66851 


.94 680 


.72 170 


.27830 


.05 320 


.33 149 


13 


48 


.66875 


.94674 


.72 201 


.27 799 


.05 326 


.33 125 


12 


49 


.66 899 


.94 667 


.72 231 


.27 769 


.05 333 


.33 101 


11 


50 


9.66 922 


9.94 660 


9.72 262 


0.27 738 


0.05 340 


0.33 078 


10 


51 


.66 946 


.94654 


.72 293 


.27 707 


.05 346 


.33 054 


9 


52 


.66970 


.94647 


.72 323 


.27 677 


.05 353 


.33 030 


8 


53 


.66994 


.94640 


.72 354 


.27646 


.05 360 


.33006 


7 


54 


.67 018 


.94 634 


.72384 


.27 616 


.05366 


.32 982 


6 


55 


9.67 042 


9.94 627 


9.72 415 


0.27 585 


0.05 373 


0.32 958 


5 


56 


.67 066 


.94 620 


.72445 


.27 555 


.05380 


.32 934 


4 


57 


.67090 


.94 614 


.72 476 


.27 524 


.05386 


.32 910 


3 


58 


.67 113 


.94607 


.72 506 


.27 494 


.05 393 


.32887 


2 


59 


.67 137 


.94600 


.72 537 


.27 463 


.05 400 


.32 863 


1 


60 


9.67 161 


9.94 593 


9.72 567 


0.27 433 


0.05 407 


0.32 839 







Cos 


Sin 


Cot 


T;in 


Csc 


Sec 


' 



117 (297) 



(242) 62 



224 



Table 4. Trigonometric Logarithms 



28 (208) 



(331) 151 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







9.67 161 


9.94 593 


9.72 567 


0.27 433 


0.05 407 


0.32 839 


60 


1 


.67 185 


.94 587 


.72 598 


.27 402 


.05413 


.32 815 


59 


2 


.67 208 


.94 580 


.72 628 


.27 372 


.05 420 


.32 792 


58 


3 


.67 232 


.94 573 


.72 659 


.27 341 


.05 427 


.32 768 


57 


4 


.67 256 


.94 567 


.72 689 


.27311 


.05 433 


.32 744 


56 


5 


9.67 280 


9.94 560 


9.72 720 


0.27 280 


0.05 440 


0.32 720 


55 


6 


.67 303 


.94 553 


.72 750 


.27 250 


.05 447 


.32 697 


54 


7 


.67 327 


.94 546 


.72 780 


.27 220 


.05 454 


.32 673 


53 


8 


.67 350 


.94 540 


.72 811 


.27 189 


.05 460 


.32 650 


52 


9 


.67 374 


.94 533 


.72841 


.27 159 


.05 467 


.32 626 


51 


10 


9.67 398 


9.94 526 


9.72 872 


0.27 128 


0.05 474 


0.32 602 


50 


11 


.67 421 


.94 519 


.72 902 


.27 098 


.05 481 


.32 579 


49 


12 


.67 445 


.94 513 


.72 932 


.27 068 


.05 487 


.32 555 


48 


13 


.67 468 


.94 506 


.72 963 


.27 037 


.05 494 


.32 532 


47 


14 


.67 492 


.94 499 


.72 993 


.27 007 


.05 501 


.32 508 


46 


15 


9.67 515 


9.94 492 


9.73 023 


0.26 977 


0.05 508 


0.32 485 


45 


16 


.67 539 


.94 485 


.73 054 


.26 946 


.05 515 


.32 461 


44 


17 


.67 562 


.94 479 


.73084 


.26 916 


.05 521 


.32 438 


43 


18 


.67 586 


.94 472 


.73 114 


.26 886 


.05 528 


.32 414 


42 


19 


.67 609 


.94 465 


.73 144 


.26 856 


.05 535 


.32 391 


41 


20 


9.67 633 


9.94 458 


9.73 175 


0.26 825 


0.05 542 


0.32 367 


40 


21 


.67 656 


.94 451 


.73 205 


.26 795 


.05 549 


.32 344 


39 


22 


.67 680 


.94 445 


.73 235 


.26 765 


.05 555 


.32 320 


38 


23 


.67 703 


.94 438 


.73 265 


.26 735 


.05 562 


.32 297 


37 


24 


.67 726 


.94 431 


.73 295 


.26 705 


.05 569 


.32 274 


36 


25 


9.67 750 


9.94 424 


9.73 326 


0.26 674 


0.05 576 


0.32 250 


35 


26 


.67 773 


.94417 


.73 356 


.26 644 


.05 583 


.32 227 


34 


27 


.67 796 


.94 410 


.73 386 


.26 614 


.05 590 


.32 204 


33 


28 


.67 820 


.94 404 


.73 416 


.26 584 


.05 596 


.32 180 


32 


29 


.67843 


.94 397 


.73 446 


.26 554 


.05 603 


.32 157 


31 


30 


9.67 866 


9.94 390 


9.73 476 


0.26 524 


0.05 610 


0.32 134 


30 


31 


.67 890 


.94 383 


.73 507 


.26 493 


.05 617 


.32 110 


29 


32 


.67 913 


.94 376 


.73 537 


.26 463 


.05 624 


.32 087 


28 


33 


.67 936 


.94 369 


.73 567 


.26 433 


.05 631 


.32 064 


27 


34 


.67 959 


.94 362 


.73 597 


.26 403 


.05 638 


.32 041 


26 


35 


9.67 982 


9.94 355 


9.73 627 


0.26 373 


0.05 645 


0.32 018 


25 


36 


.68 006 


.94 349 


.73 657 


.26 343 


.05 651 


.31 994 


24 


37 


.68 029 


.94 342 


.73 687 


.26 313 


.05 658 


.31 971 


23 


38 


.68 052 


.94 335 


.73 717 


.26 283 


.05 665 


.31 948 


22 


39 


.68 075 


.94 328 


.73 747 


.26 253 


.05 672 


.31 925 


21 


40 


9.68 098 


9.94 321 


9.73 777 


0.26 223 


0.05 679 


0.31 902 


20 


41 


.68 121 


.94 314 


.73 807 


.26 193 


.05 686 


.31 879 


19 


42 


.68 144 


.94 307 


.73 837 


.26 163 


.05 693 


.31 856 


18 


43 


.68 167 


.94 300 


.73 867 


.26 133 


.05 700 


.31 833 


17 


44 


.68 190 


.94 293 


.73 897 


.26 103 


.05 707 


.31 810 


16 


45 


9.68 213 


9.94 286 


9.73 927 


0.26 073 


0.05 714 


0.31 787 


15 


46 


.68 237 


.94 279 


.73 957 


.26 043 


.05 721 


.31 763 


14 


47 


.68 260 


.94 273 


.73 987 


.26 013 


.05 727 


.?! 740 


13 


48 


.68 283 


.94 266 


.74 017 


.25 983 


.05 734 


.31 717 


12 


49 


.68 305 


.94 259 


.74 047 


.25 953 


.05 741 


.31 695 


11 


50 


9.68 328 


9.94 252 


9.74 077 


0.25 923 


0.05 748 


0.31 672 


10 


51 


.68 351 


.94 245 


.74 107 


.25 893 


.05 755 


.31 649 


9 


52 


.68 374 


.94 238 


.74 137 


.25 863 


.05 762 


.31 626 


8 


53 


.68 397 


.94 231 


.74 166 


.25 834 


.05 769 


.31 603 


7 


54 


.68 420 


.94 224 


.74 196 


.25 804 


.05 776 


.31 580 


6 


55 


9.68 443 


9.94 217 


9.74 226 


0.25 774 


0.05 783 


0.31 557 


5 


56 


.68 466 


.94 210 


.74 256 


.25 744 


.05 790 


.31 534 


4 


57 


.68 489 


.94 203 


.74 286 


.25 714 


.05 797 


.31 511 


3 


58 


.68512 


.94 196 


.74 316 


.25 684 


.05804 


.31 488 


2 


59 


.68 534 


.94 189 


.74 345 


.25 655 


.05811 


.31 466 


1 


60 


9.68 557 


9.94 182 


9.74 375 


0.25 625 


0.05 818 


0.31 443 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



118 (298) 



(241) 61 C 



Table 4. Trigonometric Logarithms 



225 



29 (209) 



(330) 150 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 





9.68 557 


9.94 182 


9.74 375 


0.25 625 


0.05 818 


0.31 443 


60 


1 


.68 580 


.94 175 


.74 405 


.25 595 


.05 825 


.31 420 


59 


2 


.68 603 


.94 168 


.74 435 


.25 565 


.05832 


.31 397 


58 


3 


.68 625 


.94 161 


.74 465 


.25 535 


.05839 


.31 375 


57 


4 


.68648 


.94154 


.74 494 


.25506 


.05846 


.31 352 


56 


5 


9.68 671 


9.94 147 


9.74 524 


0.25 476 


0.05 853 


0.31 329 


55 


6 


.68 694 


.94 140 


.74 554 


.25446 


.05 860 


.31 306 


54 


7 


.68 716 


.94 133 


.74 583 


.25 417 


.05 867 


.31 284 


53 


8 


.68 739 


.94 126 


.74 613 


.25 387 


.05 874 


.31 261 


52 


9 


.68 762 


.94 119 


.74643 


.25 357 


.05 881 


.31 238 


51 


10 


9.68 784 


9.94 112 


9.74 673 


0.25 327 


0.05 888 


0.31 216 


50 


11 


.68807 


.94 105 


.74 702 


.25 298 


.05 895 


.31 193 


49 


12 


.68 829 


.94 098 


.74 732 


.25 268 


.05 902 


.31 171 


48 


13 


.68852 


.94 090 


.74 762 


.25 238 


.05 910 


.31 148 


47 


14 


.68 875 


.94083 


.74 791 


.25 209 


.05 917 


.31 125 


46 


15 


9.68 897 


9.94 076 


9.74 821 


0.25 179 


0.05 924 


0.31 103 


45 


16 


.68 920 


.94 069 


.74 851 


.25 149 


.05 931 


.31 080 


44 


17 


.68 942 


.94 062 


.74 880 


.25 120 


.05 938 


.31 058 


43 


18 


.68 965 


.94 055 


.74 910 


.25 090 


.05 945 


.31 035 


42 


19 


.68 987 


.94048 


.74 939 


.25 061 


.05 952 


.31 013 


41 


20 


9.69 010 


9.94 041 


9.74 969 


0.25 031 


0.05 959 


0.30 990 


40 


21 


.69 032 


.94 034 


.74 998 


.25 002 


.05 966 


.30 968 


39 


22 


.69 055 


.94 027 


.75 028 


.24 972 


.05 973 


.30 945 


38 


23 


.69 077 


.94 020 


.75 058 


.24 942 


.05 980 


.30 923 


37 


24 


.69 100 


.94 012 


.75 087 


.24 913 


.05 988 


.30 900 


36 


25 


9.69 122 


9.94 005 


9.75 117 


0.24 883 


0.05 995 


0.30 878 


35 


26 


.69 144 


.93 998 


.75 146 


.24 854 


.06002 


.30 856 


34 


27 


.69 167 


.93 991 


.75 176 


.24 824 


.06 009 


.30 833 


33 


28 


i69 189 


.93984 


.75 205 


.24 795 


.06 016 


.30811 


32 


29 


.69 212 


.93 977 


.75 235 


.24 765 


.06023 


.30 788 


31 


30 


9.69 234 


9.93 970 


9.75 264 


0.24 736 


0.06 030 


0.30 766 


30 


31 


.69 256 


.93 963 


.75 294 


.24 706 


.06 037 


.30744 


29 


32 


.69 279 


.93 955 


.75 323 


.24 677 


.06045 


.30 721 


28 


33 


.69 301 


.93948 


.75 353 


.24647 


.06 052 


.30 699 


27 


34 


.69 323 


.93 941 


.75 382 


.24618 


.06 059 


.30 677 


26 


35 


9.69 345 


9.93 934 


9.75411 


0.24 589 


0.06 066 


0.30 655 


25 


36 


.69 368 


.93 927 


.75 441 


.24 559 


.06073 


.30 632 


24 


37 


.69 390 


.93 920 


.75 470 


.24530 


.06 080 


.30 610 


23 


38 


.69 412 


.93 912 


.75500 


.24 500 


.06 088 


.30588 


22 


39 


.69 434 


.93 905 


.75 529 


.24 471 


.06 095 


.30 566 


21 


40 


9.69 456 


9.93 898 


9.75 558* 


0.24 442 


0.06 102 


0.30 544 


20 


41 


.69 479 


.93 891 


.75 588 


.24 412 


.06 109 


.30 521 


19 


42 


.69 501 


.93884 


.75 617 


.24 383 


.06 116 


.30 499 


18 


43 


.69 523 


.93 876 


.75647 


.24 353 


.06 124 


.30 477 


17 


44 


.69545 


.93 869 


.75 676 


.24 324 


.06 131 


.30 455 


16 


45 


9.69 567 


9.93 862 


9.75 705 


0.24 295 


0.06 138 


0.30 433 


15 


46 


.69 589 


.93855 


.75 735 


.24 265 


.06 145 


.30411 


14 


47 


.69611 


.93847 


.75764 


.24 236 


.06 153 


.30 389 


13 


48 


.69 633 


.93840 


.75 793 


.24 207 


.06 160 


.30 367 


12 


49 


.69 655 


..93 833 


.75 822 


.24 178 


.06167 


.30 345 


11 


50 


9.69 677 


9.93 826 


9.75 852 


0.24 148 


0.06 174 


0.30 323 


10 


51 


.69 699 


.93 819 


.75 881 


.24 119 


.06 181 


.30 301 


9 


52 


.69 721 


.93811 


.75 910 


.24 090 


.06 189 


.30 279 


8 


53 


.69 743 


.93804 


.75 939 


.24061 


.06 196 


.30 257 


7 


54 


.69 765 


.93 797 


.75 969 


.24 031 


.06 203 


.30 235 


6 


55 


9.69 787 


9.93 789 


9.75 998 


0.24 002 


0.06211 


0.30 213 


5 


56 


.69809 


.93 782 


.76 027 


.23 973 


.06 218 


.30 191 


4 


57 


.69831 


.93 775 


.76 056 


.23944 


.06 225 


.30 169 


3 


58 


.69853 


.93 768 


.76 086 


.23 914 


.06 232 


.30 147 


2 


59 


.69 875 


.93 760 


.76 115 


.23 885 


.06 240 


.30 125 


1 


60 


9.69 897 


9.93 753 9.76 144 


0.23 856 


0.06 247 


0.30 103 







Cos 


Sin Cot 


T;in 


Csc 


Sec 


' 



119 (299) 



(240) 60 



226 



Table 4. Trigonometric Logarithms 



30 (210) 



(329) 149 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







9.69 897 


9.93 753 


9.76 144 


0.23 856 


0.06 247 


0.30 103 


60 


1 


.69 919 


.93 746 


.76 173 


.23 827 


.06 254 


.30 081 


59 


2 


.69 941 


.93 738 


.76 202 


.23 798 


.06 262 


.30 059 


58 


3 


.69 963 


.93 731 


.76 231 


.23 769 


.06 269 


.30 037 


57 


4 


.69984 


.93 724 


.76 261 


.23 739 


.06 276 


.30 016 


56 


5 


9.70 006 


9.93 717 


9.76 290 


0.23 710 


0.06 283 


0.29 994 


55 


6 


.70 028 


.93 709 


.76 319 


.23 681 


.06 291 


.29 972 


54 


7 


.70 050 


.93 702 


.76 348 


.23 652 


.06 298 


.29 950 


53 


8 


.70 072 


.93 695 


.76 377 


.23 623 


.06 305 


.29 928 


52 


9 


.70 093 


.93 687 


.76 406 


.23 594 


.06 313 


.29 907 


51 


10 


9.70 115 


9.93 680 


9.76 435 


0.23 565 


0.06 320 


0.29 885 


50 


11 


.70 137 


.93 673 


.76464 


.23 536 


.06 327 


.29 863 


49 


12 


.70 159 


.93 665 


.76 493 


.23 507 


.06 335 


.29841 


48 


13 


.70 180 


.93 658 


.76 522 


.23 478 


.06 342 


.29 820 


45 


14 


.70 202 


.93 650 


.76 551 


.23 449 


.06 350 


.29 798 


46 


15 


9.70 224 


9.93 643 


9.76 580 


0.23 420 


0.06 357 


0.29 776 


45 


16 


.70 245 


.93 636 


.76 609 


.23 391 


.06 364 


.29 755 


44 


17 


.70 267 


.93 628 


.76 639 


.23 361 


.06 372 


.29 733 


43 


18 


.70 288 


.93 621 


.76 668 


.23 332 


.06 379 


.29 712 


42 


19 


.70 310 


.93 614 


.76 697 


.23 303 


.06 386 


.29 690 


41 


20 


9.70 332 


9.93 606 


9.76 725 


0.23 275 


0.06 394 


0.29 668 


40 


21 


.70 353 


.93 599 


.76 754 


.23 246 


.06 401 


.29 647 


39 


22 


.70 375 


.93 591 


.76 783 


.23 217 


.06 409 


.29 625 


38 


23 


.70 396 


.93584 


.76 812 


.23 188 


.06 416 


.29 604 


37 


24 


.70 418 


.93 577 


.76841 


.23 159 


.06 423 


.29 582 


36 


25 


9.70 439 


9.93 569 


9.76 870 


0.23 130 


0.06 431 


0.29 561 


35 


26 


.70 461 


.93 562 


.76 899 


.23 101 


.06 438 


.29 539 


34 


27 


.70 482 


.93 554 


.76 928 


.23 072 


.06 446 


.29 518 


33 


28 


.70 504 


.93 547 


.76 957 


.23 043 


.06 453 


.29 496 


32 


29 


.70 525 


.93 539 


.76 986 


.23 014 


.06 461 


.29 475 


31 


30 


9.70 547 


9.93 532 


9.77 015 


0.22 985 


0.06 468 


0.29 453 


30 


31 


.70 568 


.93 525 


.77 044 


.22 956 


.06 475 


.29 432 


29 


32 


.70 590 


.93 517 


.77 073 


.22 927 


.06 483 


.29 410 


28 


33 


.70611 


.93 510 


.77 101 


.22 899 


.06 490 


.29 389 


27 


34 


.70 633 


.93 502 


.77 130 


.22 870 


.06 498 


.29 367 


26 


35 


9.70 654 


9.93 495 


9.77 159 


0.22 841 


0.06 505 


0.29 346 


25 


36 


.70 675 


.93 487 


.77 188 


.22 812 


.06 513 


.29 325 


24 


37 


.70 697 


.93 480 


.77 217 


.22 783 


.06 520 


.29 303 


23 


38 


.70 718 


.93 472 


.77 246 


.22 754 


.06 528 


.29 282 


22 


39 


.70 739 


.93 465 


.77 274 


.22 726 


.06 535 


.29 261 


21 


40 


9.70 761 


9.93 457 


9.77 303 " 


0.22 697 


0.06 543 


0.29 239 


20 


41 


.70 782 


.93 450 


.77 332 


.22 668 


.06 550 


.29218 


19 


42 


.70 803 


.93 442 


.77 361 


.22 639 


.06 558 


.29 197 


18 


43 


.70 824 


.93 435 


.77 390 


.22 610 


.06 505 


.29 176 


17 


44 


.70 846 


.93 427 


.77 418 


.22 582 


.06 573 


.29 154 


16 


45 


9.70 867 


9.93 420 


9.77 447 


0.22 553 


0.06 580 


0.29 133 


15 


46 


.70 888 


.93 412 


.77 476 


.22 524 


.06 588 


.29 112 


14 


47 


.70 909 


.93 405 


.77 505 


.22 495 


.06 595 


29091 


13 


48 


.70 931 


.93 397 


.77 533 


.22 467 


.06 603 


.29 069 


12 


49 


.70 952 


.93 390 


.77 562 


.22 438 


.06 610 


.29 048 


11 


50 


9.70 973 


9.93 382 


9.77 591 


0.22 409 


0.06 618 


0.29 027 


10 


51 


.70 994 


.93 375 


.77 619 


.22 381 


.06 625 


.29 006 


9 


52 


.71 015 


.93 367 


.77 648 


.22 352 


.06 633 


.28 985 


8 


53 


.71 036 


.93 360 


.77 677 


.22 323 


.06 640 


.28964 


7 


54 


.71 058 


.93 352 


.77 706 


.22 294 


.06 648 


.28 942 


6 


55 


9.71 079 


9.93 344 


9.77 734 


0.22 266 


0.06 656 


0.28 921 


5 


56 


.71 100 


.93 337 


.77 763 


.22 237 


.06 663 


.28 900 


4 


57 


.71 121 


.93 329 


.77 791 


.22 209 


.06 671 


.28 879 


3 


58 


.71 142 


.93 322 


.77 820 


.22 180 


.06 678 


.28 858 


2 


59 


.71 163 


.93 314 


.77 849 


.22 151 


.06 686 


.28837 


1 


60 


9.71 184 


9.93 307 


9.77 877 


0.22 123 


0.06 693 


0.28816 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



120 (300) 



(239) 59 



Table 4. Trigonometric Logarithms 



227 



31 (211) 



(328) 148 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







9.71 184 


9.93 307 


9.77 877 


0.22 liJ3 


0.06 693 


0.28 816 


60 


1 


.71 205 


.93 299 


.77 906 


.22 094 


.06 701 


.28 795 


59 


2 


.71 226 


.93 291 


.77 935 


.22065 


.06 709 


.28 774 


58 


3 


.71 247 


.93284 


.77 963 


.22 037 


.06 716 


.28 753 


57 


4 


.71 268 


.93 276 


.77 992 


.22008 


.06724 


.28 732 


56 


5 


9.71 289 


9.93 269 


9.78 020 


0.21 980 


0.06 731 


0.28 711 


55 


6 


.71 310 


.93 261 


.78 049 


.21 951 


.06 739 


.28 690 


54 


7 


.71 331 


.93 253 


.78 077 


.21 923 


.06 747 


.28 669 


53 


8 


.71 352 


.93 246 


.78 106 


'.21 894 


.06754 


.28 648 


52 


9 


.71 373 


.93 238 


.78 135 


.21 865 


.06762 


.28 627 


51 


10 


9.71 393 


9.93 230 


9.78 163 


0.21 837 


0.06 770 


0.28 607 


50 


11 


.71 414 


.93 223 


.78 192 


.21 808 


.06 777 


.28 586 


49 


12 


.71 435 


.93 215 


.78 220 


.21 780 


.06785 


.28 565 


48 


13 


.71 456 


.93 207 


.78 249 


.21 751 


.06 793 


.28 544 


47 


14 


.71 477 


.93 200 


.78 277 


.21 723 


.06 800 


.28 523 


46 


15 


9.71 498 


9.93 192 


9.78 306 


0.21 694 


0.06 808 


0.28 502 


45 


16 


.71 519 


.93 184 


.78 334 


.21 666 


.06 816 


.28 481 


44 


17 


.71 539 


.93 177 


.78 363 


.21 637 


.06823 


.28 461 


43 


18 


.71 560 


.93 169 


.78 391 


.21 609 


.06 831 


.28440 


42 


19 


.71 581 


.93 161 


.78 419 


.21 581 


.06839 


.28 419 


41 


20 


9.71 602 


9.93 154 


.78 448 


0.21 552 


0.06 846 


0.28 398 


40 


21 


.71 622 


.93 146 


.78 476 


.21 524 


.06 854 


.28 378 


39 


22 


.71 643 


.93 138 


.78 505 


.21 495 


.06 862 


.28 357 


38 


23 


.71664 


.93 131 


.78 533 


.21 467 


.06 869 


.28 336 


37 


24 


.71 685 


.93 123 


.78 562 


.21 438 


.06877 


.28 315 


36 


25 


9.71 705 


9.93 115 


9.78 590 


0.21 410 


0.06 885 


0.28 295 


35 


26 


.71 726 


.93 108 


.78 618 


.21 382 


.06 892 


.28 274 


34 


27 


.71 747 


.93 100 


.78647 


.21 353 


.06 900 


.28 253 


33 


28 


.71 767 


.93 092 


.78 675 


.21 325 


.06 908 


.28 233 


32 


29 


.71 788 


.93084 


.78 704 


.21 296 


.06 916 


.28 212 


31 


30 


9.71 809 


9.93 077 


9.78 732 


0.21 268 


0.06 923 


0.28 191 


30 


31 


.71 829 


.93 069 


.78 760 


.21 240 


.06931 


.28 171 


29 


32 


.71850 


.93 061 


.78 789 


.21 211 


.06 939 


.28 150 


28 


33 


.71 870 


.93 053 


.78 817 


.21 183 


.06 947 


.28 130 


27 


34 


.71 891 


.93046 


.78845 


.21 155 


.06 954 


.28 109 


26 


35 


9.71911 


9.93 038 


9.78 874 


0.21 126 


0.06 962 


0.28 089 


25 


36 


.71 932 


.93 030 


.78 902 


.21 098 


.06 970 


.28 068 


24 


37 


.71 952 


.93 022 


.78 930 


.21 070 


.06 978 


.28048 


23 


38 


.71 973 


.93 014 


.78 959 


.21 041 


.06 986 


.28 027 


22 


39 


.71 994 


.93 007 


.78 987 


.21 013 


.06 993 


.28 006 


21 


40 


9.72 014 


9.92 999 


9.79 015 


0.20 985 


0.07 001 


0.27 986 


20 


41 


.72 034 


.92 991 


.79043 


.20 957 


.07009 


.27 966 


19 


42 


.72 055 


.92 983 


.79 072 


.20 928 


.07 017 


.27 945 


18 


43 


.72 075 


.92 976 


.79 100 


.20900 


.07 024 


.27 925 


17 


44 


.72 096 


.92 968 


.79 128 


.20 872 


.07 032 


.27904 


16 


45 


9.72 116 


9.92 960 


9.79 156 


0.20844 


0.07 040 


0.27 884 


15 


46 


.72 137 


.92 952 


.79 185 


.20 815 


.07 048 


.27 863 


14 


47 


.72 157 


.92944 


.79 213 


.20 787 


.07 056 


.27843 


13 


48 


.72 177 


.92 936 


.79 241 


.20 759 


.07 064 


.27 823 


12 


49 


.72 198 


.92 929 


.79 269 


.20 731 


.07 071 


.27 802 


11 


50 


9.72 218 


9.92 921 


9.79 297 


0.20 703 


0.07 079 


0.27 782 


10 


51 


.72 238 


.92 913 


.79 326 


.20 674 


.07 087 


.27 762 


9 


52 


.72 259 


.92 905 


.79 354 


.20646 


.07 095 


.27 741 


8 


53 


.72 279 


.92 897 


.79 382 


.20 618 


.07 103 


.27 721 


7 


54 


.72 299 


.92889 


.79 410 


.20 590 


.07 111 


.27 701 


6 


55 


9.72 320 


9.92 881 


9.79 438 


0.20 562 


0.07 119 


0.27 680 


5 


56 


.72 340 


.92 874 


.79 466 


.20 534 


.07 126 


.27 660 


4 


57 


.72 360 


.92 866 


.79 495 


.20 505 


.07134 


.27640 


3 


58 


.72 381 


.92 858 


.79 523 


.20 477 


.07 142 


.27 619 


2 


59 


.72 401 


.92850 


.79 551 


.20 449 


.07 150 


.27 599 


1 


60 


9.72 421 


9.92 842 


9.79 579 


0.20 421 


0.07 158 


0.27 579 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



121 (301) 



(238) 58 



228 



Table 4. Trigonometric Logarithms 



32 (212) 



(327) 147 



/ 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







9.72 421 


9.92 842 


9.79 579 


0.20 421 


0.07 158 


0.27 579 


60 


1 


.72 441 


.92 834 


.79 607 


.20 393 


.07 166 


.27 559 


59 


2 


.72 461 


.92 826 


.79 635 


.20 365 


.07 174 


.27 539 


58 


3 


.72 482 


.92 818 


.79 663 


.20 337 


.07 182 


.27 518 


57 


4 


.72 502 


.92 810 


.79 691 


.20 309 


.07 190 


.27 498 


56 


5 


9.72 522 


9.92 803 


9.79 719 


0.20 281 


0.07 197 


0.27 478 


55 


6 


.72 542 


.92 795 


.79 747 


.20 253 


.07 205 


.27 458 


54 


7 


.72 562 


.92 787 


.79 776 


.20 224 


.07 213 


.27 438 


53 


8 


.72 582 


.92 779 


.79 804 


.20 196 


.07 221 


.27 418 


52 


9 


.72 602 


.92 771 


.79 832 


.20 168 


.07 229 


.27 398 


51 


10 


9.72 622 


9.92 763 


9.79 860 


0.20 140 


0.07 237 


0.27 378 


50 


11 


.72643 


.92 755 


.79 888 


.20 112 


.07 245 


.27 357 


49 


12 


.72 663 


.92 747 


.79 916 


.20084 


.07 253 


.27 337 


48 


13 


.72 683 


.92 739 


.79 944 


.20 056 


.07 261 


.27 317 


47 


14 


.72 703 


.92 731 


.79 972 


.20 028 


.07 269 


.27 297 


46 


15 


9.72 723 


9.92 723 


9.80 000 


0.20 000 


0.07 277 


0.27 277 


45 


16 


.72 743 


.92 715 


.80 028 


.19 972 


.07 285 


.27 257 


44 


17 


.72 763 


.92 707 


.80 056 


.19 944 


.07 293 


.27 237 


43 


18 


72 783 


.92 699 


.80084 


.19916 


.07 301 


.27 217 


42 


19 


.72 803 


.92 691 


.80112 


.19 888 


.07 309 


.27 197 


41 


20 


9.72 823 


9.92 683 


9.80 140 


0.19 860 


0.07 317 


0.27 177 


40 


21 


.72843 


.92 675 


.80 168 


.19 832 


.07 325 


.27 157 


39 


22 


.72 863 


.92 667 


.80 195 


.19 805 


.07 333 


.27 137 


38 


23 


.72 883 


.92 659 


.80 223 


.19 777 


.07 341 


.27 117 


37 


24 


.72 902 


.92 651 


.80 251 


.19 749 


.07 349 


.27 098 


36 


25 


9.72 922 


9.92 643 


9.80 279 


0.19 721 


0.07 357 


0.27 078 


35 


26 


.72 942 


.92 635 


.80 307 


.19 693 


.07 365 


.27 058 


34 


27 


.72 962 


.92 627 


.80 335 


.19 665 


.07 373 


.27 038 


33 


28 


.72 982 


.92 619 


.80 363 


.19 637 


.07 381 


.27 018 


32 


29 


.73 002 


.92611 


.80391 


.19 609 


.07389 


.26 998 


31 


30 


9.73 022 


0.92 603 


9.80 419 


0.19 581 


0.07 397 


0.26 978 


30 


31 


.73 041 


.92 595 


.80 447 


.19 553 


.07 405 


.26 959 


29 


32 


73 061 


.92 587 


.80 474 


.19 526 


.07 413 


.26 939 


28 


33 


.73 081 


.92 579 


.80 502 


.19 498 


.07 421 


.26 919 


27 


34 


.73 101 


.92 571 


.80 530 


.19 470 


.07 429 


.26 899 


26 


35 


9.73 121 


9.92 563 


9.80 558 


0.19 442 


0.07 437 


0.26 879 


25 


36 


.73 140 


.92 555 


.80 586 


.19414 


.07 445 


.26 860 


24 


37 


.73 160 


.92 546 


.80 614 


.19 386 


.07 454 


.26840 


23 


38 


.73 180 


.92 538 


.80 642 


.19 358 


.07 462 


.26 820 


22 


39 


.73 200 


.92 530 


.80 669 


.19331 


.07 470 


.26 800 


21 


40 


9.73 219 


9.92 522 


9.80 697 


0.19 303 


0.07 478 


0.26 781 


20 


41 


.73 239 


.92 514 


.80 725 


.19 275 


.07 486 


.26 761 


19 


42 


.73 259 


.92 506 


.80 753 


.19 247 


.07 494 


.26 741 


18 


43 


.73 278 


.92 498 


.80781 


.19219 


.07 502 


.26 722 


17 


44 


.73 298 


.92 490 


.80 808 


.19 192 


.07 510 


.26 702 


16 


45 


9.73 318 


9.92 482 


9.80 836 


0.19 164 


0.07 518 


0.26 682 


15 


46 


.73 337 


.92 473 


.80 864 


.19 136 


.07 527 


.26 663 


14 


47 


.73 357 


.92 465 


.80 892 


.19 108 


.07 535 


.26 643 


13 


48 


.73 377 


.92 457 


.80919 


.19 081 


.07 543 


.26 623 


12 


49 


.73 396 


.92 449 


.80 947 


.19 053 


.07 551 


.26 604 


11 


50 


9.73416 


9.92 441 


9.80 975 


0.19 025 


0.07 559 


0.26 584 


10 


51 


.73 435 


.92 433 


.81 003 


.18 997 


.07 567 


.26 565 


9 


52 


.73 455 


.92 425 


.81 030 


.18970 


.07 575 


.26 545 


8 


53 


.73 474 


.92 416 


.81 058 


.18 942 


.07 584 


.26 526 


7 


54 


.73 494 


.92 408 


.81 086 


.18914 


.07 592 


.26 506 


6 


55 


9.73 513 


9.92 400 


9.81 113 


0.18887 


0.07 600 


0.26 487 


5 


56 


.73 533 


.92 392 


.81 141 


.18 859 


.07 608 


.26 467 


4 


57 


.73 552 


.92 384 


.81 169 


.18831 


.07 616 


.26 448 


3 


58 


.73 572 


.92 376 


.81 196 


.18 804 


.07 624 


.26 428 


2 


59 


.73 591 


.92 367 


.81 224 


.18 776 


.07 633 


.26 409 


1 


60 


9.73611 


9.92 359 


9.81 252 


0.18 748 


0.07 641 


0.26 389 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



122 C302) 



(237) 57 



Table 4. Trigonometric Logarithms 



229 



33 (213) 



(326) 146 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







9.73611 


9.92 359 


9.81 252 


0.18 748 


0.07 641 


0.26 389 


60 


1 


.73 630 


.92 351 


.81 279 


.18721 


.07 649 


.26 370 


59 


2 


.73 650 


.92 343 


.81 307 


.18 693 


.07 657 


.26 350 


58 


3 


.73 669 


.92 335 


.81 335 


.18 665 


.07 665 


.26 331 


57 


4 


.73 689 


.92 326 


.81 362 


.18 638 


.07 674 


.26311 


56 


5 


9.73 708 


9.92 318 


9.81 390 


0.18610 


0.07 682 


0.26 292 


55 


6 


.73 727 


.92310 


.81 418 


.18 582 


.07 690 


.26 273 


54 


7 


.73 747 


.92 302 


.81 445 


.18 555 


.07 698 


.26 253 


53 


8 


.73 766 


.92 293 


.81 473 


.18 527 


.07 707 


.26 234 


52 


9 


.73 785 


.92 285 


.81 500 


.18500 


.07 715 


.26 215 


51 


10 


9.73 805 


9.92 277 


9.81 528 


0.18472 


0.07 723 


0.26 195 ' 


50 


11 


.73 824 


.92 269 


.81 556 


.18444 


.07 731 


.26 176 


49 


12 


.73 843 


.92 260 


.81 583 


.18417 


.07 740 


.26 157 


48 


13 


.73 863 


.92 252 


.81 611 


.18 389 


.07 748 


.26 137 


47 


14 


.73 882 


.92 244 


.81 638 


.18 362 


.07 756 


.26 118 


46 


15 


9.73 901 


9.92 235 


9.81 666 


0.18 334 


0.07 765 


0.26 099 


45 


16 


.73 921 


.92 227 


.81 693 


.18307 


.07 773 


.26 079 


44 


17 


.73 940 


.92 219 


.81 721 


.18 279 


.07 781 


.26 060 


43 


18 


.73 959 


.92211 


.81 748 


.18 252 


.07 789 


.26041 


42 


19 


.73 978 


.92 202 


.81 776 


.18 224 


.07 798 


.26 022 


41 


20 


9.73 997 


9.92 194 


9.81 803 


0.18 197 


0.07 806 


0.26 003 


40 


21 


.74 017 


.92 186 


.81 831 


.18 169 


.07 814 


.25 983 


39 


22 


.74 036 


.92 177 


.81 858 


.18 142 


.07 823 


.25 964 


38 


23 


.74 055 


.92 169 


.81 886 


.18114 


.07 831 


.25 945 


37 


24 


.74 074 


.92 161 


.81 913 


.18087 


.07 839 


.25 926 


36 


25 


9.74 093 


9.92 152 


9.81 941 


0.18 059 


0.07 848 


0.25 907 


35 


26 


.74 113 


.92 144 


.81 968 


.18 032 


.07 856 


.25 887 


34 


27 


.74 132 


.92 136 


.81 996 


.18004 


.07 864 


.25 868 


33 


28 


.74 151 


.92 127 


.82 023 


.17 977 


.07 873 


.25849 


32 


29 


.74 170 


.92 119 


.82 051 


.17 949 


.07 881 


.25 830 


31 


30 


9.74 189 


9.92 111 


9.82 078 


0.17 922 


0.07 889 


0.25811 


30 


31 


.74 208 


.92 102 


.82 106 


.17 894 


.07 898 


.25 792 


29 


32 


.74 227 


.92 094 


.82 133 


.17 867 


.07 906 


.25 773 


28 


33 


.74 246 


.92 086 


.82 161 


.17 839 


.07 914 


.25 754 


27 


34 


.74 265 


.92 077 


.82 188 


.17812 


.07 923 


.25 735 


26 


35 


9.74 284' 


9.92 069 


9.82 215 


0.17 785 


0.07 931 


0.25 716 


25 


36 


.74 303 


.92 060 


.82 243 


.17 757 


.07 940 


.25 697 


24 


37 


.74 322 


.92 052 


.82 270 


.17 730 


.07 948 


.25 678 


23 


38 


.74 341 


.92 044 


.82 298 


.17 702 


.07 956 


.25 659 


22 


39 


.74 360 


.92 035 


.82 325 


.17 675 


.07 965 


.25640 


21 


40 


9.74 379 


9.92 027 


9.82 352 


0.17 648 


0.07 973 


0.25 621 


20 


41 


.74 398 


.92 018 


.82 380 


.17 620 


.07 982 


.25 602 


19 


42 


.74 417 


.92 010 


.82 407 


.17 593 


.07 990 


.25 583 


18 


43 


.74 436 


.92 002 


.82 435 


.17 565 


.07 998 


.25564 


17 


44 


.74 455 


.91 993 


.82 462 


.17 538 


.08 007 


.25 545 


16 


45 


9.74 474 


9.91 985 


9.82 489 


0.17511 


0.08 015 


0.25 526 


15 


46 


.74 493 


.91 976 


.82 517 


.17483 


.08 024 


.25 507 


14 


47 


.74 512 


.91 968 


.82 544 


.17 456 


.08 032 


.25 488 


13 


48 


.74 531 


.91 959 


.82 571 


.17 429 


.08041 


.25 469 


12 


49 


.74 549 


.91 951 


.82 599 


.17401 


.08049 


.25 451 


11 


50 


9.74 568 


9.91 942 


9.82 626 


0.17 374 


0.08 058 


0.25 432 


10 


51 


.74 587 


.91 934 


.82 653 


.17 347 


.08 066 


.25 413 


9 


52 


.74 606 


.91 925- 


.82 681 


.17319 


.08 075 


.25 394 


8 


53 


.74 625 


.91 917 


.82 708 


.17 292 


.08 083 


.25375 


7 


54 


.74 644 


.91 908 


.82 735 


.17 265 


.08 092 


.25 356 


6 


55 


9.74 662 


9.91 900 


9.82 762 


0.17 238 


0.08 100 


0.25 338 


5 


56 


.74 681 


.91 891 


.82 790 


.17210 


.08 109 


.25 319 


4 


57 


.74 700 


.91 883 


.82 817 


.17183 


.08117 


.25 300 


3 


58 


.74 719 


.91 874 


.82844 


.17 156 


.08 126 


.25 281 


2 


59 


.74 737 


.91 866 


.82 871 


.17129 


.08 134 


.25 263 


1 


60 


9.74 756 


9.91 857 


9.82 899 


0.17 101 


0.08 143 


0.25 244 







Cos 


Sin 


Cot 


Tan Csc 


Sec 


' 



123 (303) 



(236) 56 



230 



Table 4. Trigonometric Logarithms 



34 (214) 



(325) 145 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







9.74 756 


9.91 857 


9.82 899 


0.17 101 


0.08 143 


0.25 244 


60 


1 


.74 775 


.91849 


.82 926 


.17 074 


.08 151 


.25 225 


59 


2 


.74 794 


.91840 


.82 953 


.17 047 


.08 160 


.25 206 


58 


3 


.74 812 


.91 832 


.82 980 


.17 020 


.08 168 


.25 188 


57 


4 


.74 831 


.91 823 


.83008 


.16 992 


.08 177 


.25 169 


56 


5 


9.74 850 


9.91 815 


9.83 035 


0.16 965 


0.08 185 


0.25 150 


55 


6 


.74 868 


.91 806 


.83 062 


.16 938 


.08 194 


.25 132 


54 


7 


.74 887 


.91 798 


.83 089 


.16911 


.08 202 


.25 113 


53 


8 


.74 906 


.91 789 


.83 117 


.16 883 


.08211 


.25 094 


52 


9 


.74 924 


.91 781 


.83 144 


.16 856 


.08 219 


.25 076 


51 


10 


9.74 943 


9.91 772 


9.83 171 


0.16 829 


0.08 228 


0.25 057 


50 


11 


.74 961 


.91 763 


.83 198 


.16 802 


.08 237 


.25 039 


49 


12 


.74 980 


.91 755 


.83225 


.16 775 


.08 245 


.25 020 


48 


13 


.74 999 


.91 746 


.83252 


.16 748 


.08 254 


.25 001 


47 


14 


.75 017 


.91 738 


.83280 


16 720 


.08 262 


.24 983 


46 


15 


9.75 036 


9.91 729 


9.83 307 


0.16 693 


0.08 271 


0.24 964 


45 


16 


.75 054 


.91 720 


.83 334 


.16 666 


.08 280 


.24 946 


44 


17 


.75 073 


.91 712 


.83361 


.16 639 


.08 288 


.24 927 


43 


18 


.75 091 


.91 703 


.83 388 


.16612 


.08 297 


.24 909 


42 


19 


.75 110 


.91 695 


.83415 


.16 585 


.08 305 


.24 890 


41 


20 


9.75 128 


9.91 686 


9.83 442 


0.16 558 


0.08 314 


0.24 872 


40 


21 


.75 147 


.91 677 


.83470 


.16 530 


.08 323 


.24 853 


39 


22 


.75 165 


.91 669 


.83 497 


.16 503 


.08 331 


.24 835 


38 


23 


.75184 


.91 660 


.83524 


.16476 


.08 340 


.24 816 


37 


24 


.75 202 


.91 651 


.83551 


.16449 


.08 349 


.24 798 


36 


25 


9.75 221 


9.91 643 


9.83 578 


0.16 422 


0.08 357 


0.24 779 


35 


26 


.75 239 


.91 634 


.83 605 


.16 395 


.08 366 


.24 761 


34 


27 


.75 258 


.91 625 


.83632 


.16 368 


.08 375 


.24 742 


33 


28 


.75 276 


.91 617 


.83659 


.16 341 


.08 383 


.24 724 


32 


29 


.75 294 


.91 608 


.83686 


.16314 


.08 392 


.24 706 


31 


30 


9.75 313 


9.91 599 


9.83 713 


0.16 287 


0.08 401 


0.24 687 


30 


31 


.75 331 


.91 591 


.83740 


.16 260 


.08 409 


.24 669 


29 


32 


.75 350 


.91 582 


.83 768 


.16 232 


.08 418 


.24 650 


28 


33 


.75 368 


.91 573 


.83795 


.16 205 


.08 427 


.24 632 


27 


34 


.75 386 


.91 565 


.83822 


.16 178 


.08 435 


.24 614 


26 


35 


9.75 405 


9.91 556 


9.83 849 


0.16 151 


0.08 444 


0.24 595 


25 


36 


.75 423 


.91 547 


.83 876 


.16 124 


.08 453 


.24 577 


24 


37 


.75 441 


.91 538 


.83903 


.16 097 


.08 462 


.24 559 


23 


38 


.75 459 


.91 530 


.83930 


.16 070 


.08 470 


.24 541 


22 


39 


.75 478 


.91 521 


.83 957 


.16043 


.08 479 


.24 522 


21 


40 


9.75 496 


9.91 512 


9.83 984 


0.16016 


0.08 488 


24 504 


20 


41 


.75 514 


.91504 


.84011 


.15 989 


.08 496 


.24 486 


19 


42 


.75 533 


.91 495 


.84038 


.15 962 


.08 505 


.24 467 


18 


43 


.75 551 


.91 486 


.84065 


.15 935 


.08 514 


.24 449 


17 


44 


.75 569 


.91 477 


.84092 


.15 908 


.08 523 


.24 431 


16 


45 


9.75 587 


9.91 469 


9.84 119 


0.15 881 


0.08 531 


0.24 413 


15 


46 


.75 605 


.91 460 


.84146 


.15 854 


.08 540 


.24 395 


14 


47 


.75 624 


.91 451 


.84173 


.15 827 


.08 549 


.24 376 


13 


48 


.75642 


.91 442 


.84200 


.15 800 


.08 558 


.24 358 


12 


49 


.75 660 


.91 433 


.84227 


.15 773 


.08 567 


.24 340 


11 


50 


9.75 678 


9.91 425 


9.84 254 


0.15 746 


0.08 575 


0.24 322 


10 


51 


.75 696 


.91 416 


.84 280 


.15 720 


.08584 


.24 304 


9 


52 


.75 714 


.91 407 


.84307 


.15 693- 


.08 593 


.24 286 


8 


53 


.75 733 


.91 398 


.84334 


.15 666 


.08 602 


.24 267 


7 


54 


.75 751 


.91 389 


.84361 


.15 639 


.08611 


.24 249 


6 


55 


9.75 769 


9.91 381 


9.84 388 


0.15 612 


0.08 619 


0.24 231 


5 


56 


.75 787 


.91 372 


.84415 


.15 585 


.08 628 


.24 213 


4 


57 


.75 805 


.91 363 


.84442 


.15 558 


.08 637 


.24 195 


3 


58 


.75 823 


.91 354 


.84469 


.15 531 


.08 646 


.24 177 


2 


59 


.75841 


.91 345 


.84496 


.15 504 


.08 665 


.24 159 


1 


60 


9.75 859 


9.91 336 


9.84 523 


0.15 477 


0.08 664 


0.24 141 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



124 (304) 



(235) 55 



Table 4. Trigonometric Logarithms 



231 



35 (215) 



(324) 144 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







9.75 859 


9.91 336 


9.84 523 


0.15477 


0.08 664 


0.24 141 


60 


1 


.75 877 


.91 328 


.84550 


.15 450 


.08 672 


.24 123 


59 


2 


.75 895 


.91 319 


.84576 


.15 424 


.08 681 


.24 105 


58 


3 


.75 913 


.91 310 


.84603 


.15 397 


.08 690 


.24087 


57 


4 


.75 931 


.91 301 


.84630 


.15 370 


.08 699 


.24 069 


56 


5 


9.75 949 


9.91 292 


9.84 657 


0.15 343 


0.08 708 


0.24 051 


55 


6 


.75 967 


.91 283 


.84684 


.15316 


.08 717 


.24 033 


54 


7 


.75985 


.91 274 


.84711 


.15 289 


.08 726 


.24 015 


53 


8 


.76003 


.91 266 


.84738 


.15 262 


.08 734 


.23 997 


52 


9 


.76 021 


.91 257 


.84764 


.15 236 


.08 743 


.23 979 


51 


10 


9.76 039 


9.91 248 


9.84 791 


0.15209 


0.08 752 


0.23 961 


50 


11 


.76 057 


.91 239 


.84818 


.15 182 


.08 761 


.23 943 


49 


12 


.76 075 


.91 230 


.84845 


.15 155 


.08 770 


.23 925 


48 


13 


.76 093 


.91 221 


.84872 


.15 128 


.08 779 


.23 907 


47 


14 


.76111 


.91 212 


.84899 


.15 101 


.08 788 


.23 889 


46 


15 


9.76 129 


9.91 203 


9.84 925 


0.15 075 


0.08 797 


0.23 871 


45 


16 


.76 146 


.91 194 


.84952 


.15 048 


.08 806 


.23 854 


44 


17 


.76164 


.91 185 


.84979 


.15 021 


.08 815 


.23 836 


43 


18 


.76 182 


.91 176 


.85 006 


.14 994 


.08 824 


.23 818 


42 


19 


.76 200 


.91 167 


.85033 


.14 967 


.08 833 


.23800 


41 


20 


9.76 218 


9.91 158 


9.85 059 


0.14 941 


0.08 842 


0.23 782 


40 


21 


.76 236 


.91 149 


.85086 


.14914 


.08851 


.23764 


39 


22 


.76 253 


.91 141 


.85113 


.14 887 


.08 859 


.23 747 


38 


23 


.76 271 


.91 132 


.85 140 


.14 860 


.08 868 


.23 729 


37 


24 


.76 289 


.91 123 


.85 166 


.14 834 


.08 877 


.23 711 


36 


25 


9.76 307 


9.91 114 


9.85 193 


0.14 807 


0.08 886 


0.23 693 


35 


26 


.76 324 


.91 105 


.85 220 


.14 780 


.08 895 


.23 676 


34 


27 


.76 342 


.91 096 


.85 247 


.14 753 


.08 904 


.23 658 


33 


28 


.76 360 


.91 087 


.85 273 


.14 727 


.08 913 


.23 640 


32 


29 


.76 378 


.91 078 


.85 300 


.14 700 


.08 922 


.23 622 


31 


30 


9.76 395 


9.91 069 


9.85 327 


0.14 673 


0.08 931 


0.23 605 


30 


31 


.76 413 


.91 060 


.85 354 


.14646 


.08 940 


.23 587 


29 


32 


.76 431 


.91 051 


.85 380 


.14 620 


.08 949 


.23 569 


28 


33 


.76 448 


.91 042 


.85 407 


.14 593 


.08 958 


.23 552 


27 


34 


.76 466 


.91 033 


.85 434 


.14 566 


.08 967 


.23 534 


26 


35 


9.76 484 


9.91 023 


9.85 460 


0.14 540 


0.08 977 


0.23 516 


25 


36 


.76 501 


.91 014 


.85 487 


.14513 


.08 986 


.23 499 


24 


37 


.76 519 


.91 005 


.85 514 


.14 486 


.08 995 


.23 481 


23 


38 


.76 537 


.90 996 


.85 540 


.14 460 


.09004 


.23 463 


22 


39 


.76 554 


.90 987 


.85567 


.14433 


.09 013 


.23446 


21 


40 


9.76 572 


9.90 978 


9.85 594 


0.14 406 


0.09 022 


0.23 428 


20 


41 


.76 590 


.90 969 


.85620 


.14 380 


.09 031 


.23 410 


19 


42 


.76 607 


.90 960 


.85647 


.14 353 


.09040 


.23 393 


18 


43 


.76 625 


.90 951 


.85674 


.14 326 


.09049 


.23 375 


17 


44 


.76642 


.90 942 


.85700 


.14 300 


.09 058 


.23 358 


16 


45 


9.76 660 


9.90 933 


9.85 727 


0.14 273 


0.09 067 


0.23 340 


15 


46 


.76 677 


.90 924 


.85 754 


.14 246 


.09 076 


.23 323 


14 


47 


.76 695 


.90 915 


.85780 


. .14 220 


.09 085 


.23 305 


13 


48 


.76 712 


.90 906 


.85807 


.14 193 


.09 094 


.23288 


12 


49 


.76 730 


.90 896 


.85834 


.14 166 


.09104 


.23 270 


11 


50 


9.76 747 


9.90 887 


9.85 860 


0.14 140 


0.09 113 


0.23 253 


10 


51 


.76 765 


.90 878 


.85887 


.14 113 


.09 122 


.23 235 


9 


52 


.76 782 


.90 869 


.85913 


.14 087 


.09 131 


.23 218 


8 


53 


.76800 


.90860 


.85940 


.14 060 


.09 140 


.23 200 


7 


54 


.76 817 


.90851 


.85967 


.14 033 


.09 149 


.23 183 


6 


55 


9.76 835 


9.90 842 


9.85 993 


0.14 007 


0.09 158 


0.23 165 


5 


56 


.76852 


.90 832 


.86 020 


.13 980 


.09 168 


.23 148 


4 


57 


.76 870 


.90 823 


.86046 


.13 954 


.09 177 


.23 130 


3 


58 


.76 887 


.90 814 


.86 073 


.13 927 


.09186 


.23 113 


2 


59 


.76904 


.90 805 


.86100 


.13900 


.09 195 


.23 096 


1 


60 


9.76 922 


9.90 796 


9.86 126 


0.13 874 


0.09 204 


0.23 078 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



125 (305) 



(234) 54 



232 



Table 4. Trigonometric Logarithms 



36 (216) 



(323) 143 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


('so 







9.76 922 


9.90 796 


9.86 126 


0.13 874 


0.09 204 


0.23 078 


60 


1 


.76 939 


.90 787 


.86 153 


.13847 


.09 213 


.23 061 


59 


2 


.76 957 


.90 777 


.86 179 


.13 821 


.09 223 


.23 043 


58 


3 


.76 974 


.90 768 


.86 206 


.13 794 


.09 232 


.23 026 


57 


4 


.76 991 


.90 759 


.86 232 


.13 768 


.09 241 


.23 009 


56 


5 


9.77 009 


9.90 750 


9.86 259 


0.13 741 


0.09 250 


0.22 991 


55 


6 


.77 026 


.90 741 


.86 285 


.13715 


.09 259 


.22 974 


54 


7 


.77 043 


.90 731 


.86 312 


.13 688 


.09 269 


.22 957 


53 


8 


.77 061 


.90 722 


.86 338 


.13 662 


.09 278 


.22 939 


52 


9 


.77 078 


.90 713 


.86 365 


.13 635 


.09 287 


.22 922 


51 


10 


9.77 095 


9.90 704 


9.86 392 


0.13 608 


0.09 296 


0.22 905 


50 


11 


.77112 


.90 694 


.86 418 


.13 582 


.09 306 


.22 888 


49 


12 


.77 130 


.90 685 


.86 445 


.13 555 


.09 315 


.22 870 


48 


13 


.77 147 


.90 676 


.86 471 


.13 529 


.09 324 


.22 853 


47 


14 


.77 164 


.90 667 


.86 498 


.13 502 


.09 333 


.22 836 


46 


15 


9.77 181 


9.90 657 


9.86 524 


0.13 476 


0.09 343 


0.22 819 


45 


16 


.77 199 


.90648 


.86 551 


.13 449 


.09 352 


.22 801 


44 


17 


.77 216 


.90 639 


.86 577 


.13 423 


.09 361 


.22784 


43 


18 


.77 233 


.90 630 


.86 603 


.13 397 


.09 370 


.22 767 


42 


19 


.77 250 


.90 620 


.86 630 


.13 370 


.09 380 


.22 750 


41 


20 


9.77 268 


9.90611 


9.86 656 


0.13 344 


0.09 389 


0.22 732 


40 


21 


.77 285 


.90 602 


.86 683 


.13317 


.09 398 


.22 715 


39 


22 


.77 302 


.90 592 


.86 709 


.13 291 


.09 408 


.22 698 


38 


23 


.77 319 


.90 583 


.86 736 


.13 264 


.09 417 


.22 681 


37 


24 


.77 336 


.90 574 


.86762 


.13 238 


.09 426 


.22 664 


36 


25 


9.77 353 


9.90 565 


9.86 789 


0.13211 


0.09 435 


0.22 647 


35 


26 


.77 370 


.90 555 


.86 815 


.13 185 


.09 445 


.22 630 


34 


27 


.77 387 


.90 546 


.86842 


.13 158 


.09 454 


.22 613 


33 


28 


.77 405 


.90 537 


.86 868 


.13 132 


.09 463 


.22 595 


32 


29 


.77 422 


.90 527 


.86 894 


13 106 


.09 473 


.22 578 


31 


30 


9.77 439 


9.90 518 


9.86 921 


0.13 079 


0.09 482 


0.22 561 


30 


' 31 


.77 456 


.90 509 


.86 947 


.13 053 


.09 491 


.22 544 


29 


32 


.77 473 


.90 499 


.86 974 


.13 026 


.09 501 


.22 527 


28 


33 


.77 490 


.90 490 


.87000 


.13 000 


.09 510 


.22 510 


27 


34 


.77 507 


.90 480 


.87 027 


.12 973 


.09 520 


.22 493 


26 


35 


9.77 524 


9.90 471 


9.87 053 


0.12 947 


0.09 529 


0.22 476 


25 


36 


.77 541 


.90 462 


.87 079 


.12 921 


.09 538 


.22 459 


24 


37 


.77 558 


.90 452 


.87 106 


.12 894 


.09 548 


.22 442 


23 


38 


.77 575 


.90 443 


.87 132 


.12 868 


.09 557 


.22 425 


22 


39 


.77 592 


.90 434 


.87 158 


.12842 


.09 566 


.22 408 


21 


40 


9.77 609 


9.90 424 


9.87 185 


0.12815 


0.09 576 


0.22 391 


20 


41 


.77 626 


.90 415 


.87211 


.12 789 


.09 585 


.22 374 


19 


42 


.77 643 


.90 405 


.87 238 


.12 762 


.09 595 


.22 357 


18 


43 


.77 660 


.90 396 


.87 264 


.12 736 


.09 604 


.22 340 


17 


44 


.77 677 


.90 386 


.87 290 


.12710 


.09 614 


.22 323 


16 


45 


9.77 694 


9.90 377 


9.87 317 


0.12683 


0.09 623 


0.22 306 


15 


46 


.77711 


.90 368 


.87 343 


.12 657 


.09 632 


.22 289 


14 


47 


.77 728 


.90 358 


.87 369 


.12 631 


.09 642 


22272 


13 


48 


.77 744 


.90 349 


.87 396 


.12604 


.09 651 


.22 256 


12 


49 


.77 761 


.90 339 


.87 422 


.12 578 


.09 661 


.22 239 


11 


50 


9.77 778 


9.90 330 


9.87 448 


0.12 552 


0.09 670 


0.22 222 


10 


51 


.77 795 


.90 320 


.87 475 


.12 525 


.09 680 


.22 205 


9 


52 


.77 812 


.90311 


.87 501 


.12 499 


.09 689 


.22 188 


8 


53 


.77 829 


.90 301 


.87 527 


.12 473 


.09 699 


.22 171 


7 


54 


.77846 


.90 292 


.87 554 


.12 446 


.09 708 


.22 154 


6 


55 


9.77 862 


9.90 282 


9.87 580 


0.12 420 


0.09 718 


0.22 138 


5 


56 


.77 879 


.90 273 


.87 606 


.12 394 


.09 727 


.22 121 


4 


57 


.77 896 


.90 263 


.87 633 


.12 367 


'.09737 


.22 104 


3 


58 


.77 913 


.90 254 


.87 659 


.12 341 


.09 746 


.22 087 


2 


59 


.77 930 


.90 244 


.87 685 


.12315 


.09756 


.22 070 


1 


60 


9.77 946 


9.90 235 


9.87711 


0.12 289 


0.09 765 


0.22 054 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



126 (306) 



(233) 53 



Table 4. Trigonometric Logarithms 



233 



37 (217) 



(322) 142 c 



' 


Sin 


Cos 


Tan 


Cot 


Sec Csc 







9.77 946 


9.90 235 


9.87711 


0.12289 


0.09 765 


0.22 054 


60 


1 


.77 963 


.90 225 


.87 738 


.12 262 


.09 775 


.22 037 


59 


2 


.77 980 


.90 216 


.87 764 


.12 236 


.09 784 


.22 020 


58 


3 


.77 997 


.90 206 


.87 790 


.12210 


.09794 


.22 003 


57 


4 


.78 013 


.90 197 


.87 817 


.12 183 


.09803 


.21 987 


56 


5 


9.78 030 


9.90 187 


9.87 843 


0.12 157 


0.09 813 


0.21 970 


55 


6 


.78047 


.90 178 


.87 869 


.12 131 


.09 822 


.21 953 


54 


7 


.78063 


.90 168 


.87 895 


.12 105 


.09 832 


.21 937 


53 


8 


.78 080 


.90 159 


.87 922 


.12 078 


.09841 


.21 920 


52 


9 


.78 097 


.90 149 


.87 948 


.12 052 


.09 851 


.21 903 


51 


10 


9.78 113 


9.90 139 


9.87 974 


0.12 026 


0.09 861 


0.21 887 


50 


11 


.78 130 


.90 130 


.88 000 


.12 000 


.09 870 


.21 870 


49 


12 


.78 147 


.90 120 


.88 027 


.11 973 


.09 880 


.21 853 


48 


13 


.78 163 


.90111 


.88 053 


.11 947 


.09 889 


.21 837 


47 


14 


.78 180 


.90 101 


.88 079 


.11 921 


.09 899 


.21 820 


46 


15 


9.78 197 


9.90 091 


9.88 105 


0.11 895 


0.09 909 


0.21 803 


45 


16 


.78 213 


.90 082 


.88131 


.11 869 


.09 918 


.21 787 


44 


17 


.78 230 


.90 072 


.88 158 


.11 842 


.09 928 


.21 770 


43 


18 


.78 246 


.90 063 


.88184 


.11 816 


.09 937 


.21 754 


42 


19 


.78 263 


.90 053 


.88 210 


.11 790 


.09 947 


.21 737 


41 


20 


9.78 280 


9.90 043 


9.88 236 


0.11 764 


0.09 957 


0.21 720 


40 


21 


.78 296 


.90 034 


.88 262 . 


.11 738 


.09 966 


.21 704 


39 


22 


.78313 


.90 024 


.88289 


.11 711 


.09 976 


.21 687 


38 


23 


.78 329 


.90 014 


.88315 


.11 685 


.09 986 


.21 671 


37 


24 


.78 346 


.90 005 


.88341 


.11659 


.09 995 


.21 654 


36 


25 


9.78 362 


9.89 995 


9.88 367 


0.11633 


0.10 005 


0.21 638 


35 


26 


.78 379 


.89 985 


.88 393 


.11607 


.10015 


.21 621 


34 


27 


.78 395 


.89 976 


.88420 


.11580 


.10 024 


.21 605 


33 


28 


.78412 


.89 966 


.88 446 


.11 554 


.10 034 


.21588 


32 


29 


.78 428 


.89 956 


.88472 


.11528 


.10 044 


.21 572 


31 


30 


9.78 445 


9.89 947 


9.88 498 


0.11 502 


0.10 053 


0.21 555 


30 


31 


.78 461 


.89 937 


.88524 


.11476 


.10 063 


.21 539 


29 


32 


.78 478 


.89 927 


.88 550 


.11 450 


.10 073 


.21 522 


28 


33 


.78 494 


.89 918 


.88 577 


.11423 


.10 082 


.21 506 


27 


34 


.78510 


.89 908 


.88 603 


.11 397 


.10 092 


.21 490 


26 


35 


9.78 527 


9.89 898 


9.88 629 


0.11 371 


0.10 102 


0.21 473 


25 


36 


.78 543 


.89 888 


.88 655 


.11 345 


.10112 


.21 457 


24 


37 


.78 560 


.89 879 


.88 681 


.11319 


.10121 


.21 440 


23 


38 


.78 576 


.89 869 


.88 707 


.11 293 


.10 131 


.21 424 


22 


39 


.78 592 


.89859 


.88733 


.11 267 


.10 141 


.21 408 


21 


40 


9.78 609 


9.89 849 


9.88 759 


0.11 241 


0.10 151 


0.21 391 


20 


41 


.78 625 


.89840 


.88 786 


.11 214 


.10 160 


.21 375 


19 


42 


.78 642 


.89830 


.88812 


.11 188 


.10 170 


.21 358 


18 


43 


.78 658 


.89 820 


.88838 


.11 162 


.10 180 


.21 342 


17 


44 


.78 674 


.89 810 


.88864 


.11 136 


.10 190 


.21 326 


16 


45 


9.78 691 


9.89 801 


9.88 890 


0.11 110 


0.10 199 


0.21 309 


15 


46 


.78 707 


.89 791 


.88916 


.11 084 


.10 209 


.21 293 


14 


47 


.78 723 


.89 781 


.88 942 


.11058 


.10219 


.21 277 


13 


48 


.78 739 


.89 771 


.88968 


.11032 


.10 229 


.21 261 


12 


49 


.78 756 


.89 761 


.88 994 


.11 006 


.10 239 


.21 244 


11 


50 


9.78 772 


9.89 752 


9.89 020 


0.10980 


0.10 248 


0.21 228 


10 


51 


.78 788 


.89 742 


.89046 


.10 954 


.10258 


.21 212 


9 


52 


.78805 


.89 732 


.89 073 


.10 927 


.10 268 


.21 195 


8 


53 


.78 821 


.89 722 


.89 099 


.10901 


.10 278 


.21 179 


7 


54 


.78837 


.89 712 


.89 125 


.10 875 


.10 288 


.21 163 


6 


55 


9.78 853 


9.89 702 


9.89 151 


0.10 849 


0.10 298 


0.21 147 


5 


56 


.78 869 


.89 693 


.89 177 


.10 823 


.10 307 


.21 131 


4 


57 


.78886 


.89683 


.89 203 


.10 797 


.10317 


.21 114 


3 


58 


.78 902 


.89 673 


.89 229 


.10771 


.10 327 


.21 098 


2 


59 


.78 918 


.89 663 


.89 255 


.10745 


.10 337 


.21 082 


1 


60 


9.78 934 


9.89 653 


9.89 281 


0.10719 


0.10 347 


0.21 066 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



127 (307) 



(232) 52 



234 



Table 4. Trigonometric Logarithms 



38 (218) 



(321) 141 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







9.78 934 


9.89 653 


9.89 281 


0.10719 


0.10 347 


0.21 066 


60 


1 


.78 950 


.89 643 


.89 307 


.10 693 


.10 357 


.21 050 


59 


2 


.78 967 


.89 633 


.89 333 


.10 667 


.10367 


.21 033 


58 


3 


.78 983 


.89 624 


.89 359 


.10641 


.10 376 


.21 017 


57 


4 


.78 999 


.89 614 


.89 385 


.10615 


.10 386 


.21 001 


56 


5 


9.79 015 


9.89 604 


9.89411 


0.10 589 


0.10 396 


0.20 985 


55 


6 


.79 031 


.89 594 


.89 437 


.10 563 


.10 406 


.20 969 


54 


7 


.79047 


.89 584 


.89 463 


.10 537 


.10416 


.20 953 


53 


8 


.79 063 


.89 574 


.89 489 


.10511 


.10426 


.20 937 


52 


9 


.79 079 


.89 564 


.89 515 


.10485 


.10 436 


.20 921 


51 


10 


9.79 095 


9.89 554 


9.89 541 


0.10459 


0.10446 


0.20 905 


50 


11 


.79111 


.89 544 


.89 567 


.10433 


.10 456 


.20 889 


49 


12 


.79 128 


.89 534 


.89 593 


.10407 


.10466 


.20 872 


48 


13 


.79 144 


.89 524 


.89 619 


.10381 


.10476 


.20 856 


47 


14 


.79 160 


.89 514 


.89645 


.10 355 


.10 486 


.20840 


46 


15 


9.79 176 


9.89 504 


9.89 671 


0.10 329 


0.10496 


0.20 824 


45 


16 


.79 192 


.89 495 


.89 697 


.10 303 


.10 505 


.20 808 


44 


17 


.79 208 


.89 485 


.89 723 


.10 277 


.10515 


.20 792 


43 


18 


.79 224 


.89 475 


.89 749 


.10251 


.10 525 


.20 776 


42 


19 


.79 240 


.89 465 


.89 775 


.10 225 


.10 535 


.20 760 


41 


20 


9.79 256 


9.89 455 


9.89 801 


0.10 199 


0.10545 


0.20 744 


40 


21 


.79 272 


.89 445 


.89 827 


.10 173 


.10 555 


.20 728 


39 


22 


.79 288 


.89 435 


.89 853 


.10 147 


.10 565 


.20712 


38 


23 


.79 304 


.89 425 


.89 879 


.10 121 


.10 575 


.20 696 


37 


24 


.79 319 


.89 415 


.89 905 


.10 095 


.10 585 


.20 681 


36 


25 


9.79 335 


9.89 405 


9.89 931 


0.10 069 


0.10 595 


0.20 665 


35 


26 


.79 351 


.89 395 


.89 957 


.10043 


.10 605 


.20 649 


34 


27 


.79 367 


.89 385 


.89 983 


.10017 


.10615 


.20 633 


33 


28 


.79 383 


.89 375 


.90 009 


.09 991 


.10 625 


.20 617 


32 


29 


.79 399 


.89364 


.90 035 


.09 965 


.10 636 


.20 601 


31 


30 


9.79 415 


9.89 354 


9.90 061 


0.09 939 


0.10 646 


0.20 585 


30 


31 


.79 431 


.89 344 


.90 086 


.09 914 


.10 656 


.20 569 


29 


32 


.79447 


.89 334 


.90 112 


.09 888 


.10 666 


.20 553 


28 


33 


.79 463 


.89 324 


.90 138 


.09 862 


.10 676 


.20 537 


27 


34 


.79 478 


.89 314 


.90164 


.09 836 


.10 686 


.20 522 


26 


35 


9.79 494 


9.89 304 


9.90 190 


0.09 810 


0.10 696 


0.20 506 


25 


36 


.79 510 


.89 294 


.90 216 


.09784 


.10 706 


.20490 


24 


37 


.79 526 


.89284 


.90 242 


.09 758 


.10716 


.20 474 


23 


38 


.79 542 


.89 274 


.90 268 


.09 732 


.10 726 


.20 458 


22 


39 


.79 558 


.89 264 


.90 294 


.09 706 


.10 736 


.20 442 


21 


40 


9.79 573 


9.89 254 


9.90 320 


0.09 680 


O'lO 746 


0.20 427 


20 


41 


.79 589 


.89 244 


.90 346 


.09 654 


.10 756 


.20411 


19 


42 


.79 605 


.89 233 


.90 371 


.09 629 


.10 767 


.20 395 


18 


43 


.79 621 


.89 223 


.90 397 


.09 603 


.10 777 


.20 379 


17 


44 


.79 636 


.89 213 


.90 423 


.09 577 


.10 787 


.20 364 


16 


45 


9.79 652 


9.89 203 


9.90 449 


0.09 551 


0.10797 


0.20 348 


15 


46 


.79 668 


.89 193 


.90 475 


.09 525 


.10 807 


.20 332 


14 


47 


.79684 


.89 183 


.90 501 


.09 499 


.10817 


.20316 


13 


48 


.79 699 


.89 173 


.90 527 


.09 473 


.10 827 


.20 301 


12 


49 


.79 715 


.89 162 


.90 553 


.09 447 


.10 838 


.20 285 


11 


50 


9.79 731 


9.89 152 


9.90 578 


0.09 422 


0.10 848 


0.20 269 


10 


51 


.79 746 


.89 142 


.90604 


.09 396 


.10 858 


.20 254 


9 


52 


.79 762 


.89 132 


.90 630 


.09 370 


.10 868 


.20 238 


8 


53 


.79 778 


.89 122 


.90 656 


.09 344 


.10 878 


.20 222 


7 


54 


.79 793 


.89 112 


.90 682 


.09 318 


.10 888 


.20 207 


6 


55 


9.79 809 


9.89 101 


9.90 708 


0.09 292 


0.10 899 


0.20 191 


5 


56 


.79 825 


.89 091 


.90 734 


.09 266 


.10 909 


.20 175 


4 


57 


.79840 


.89 081 


.90 759 


.09 241 


.10919 


.20 160 


3 


58 


.79 856 


.89 071 


.90 785 


.09 215 


.10 929 


.20 144 


2 


59 


.79 872 


.89 060 


.90811 


.09 189 


.10 940 


.20 128 


1 


60 


9.79 887 


9.89 050 


9.90 837 


0.09 163 


0.10 950 


0.20 113 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



128 (308) 



(231) 51 



Table 4. Trigonometric Logarithms 



235 



39 (219) 



(320) 140 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







9.79 887 


9.89 050 


9.90 837 


0.09 163 


0.10950 


0.20 113 


60 


1 


.79 903 


.89 040 


.90 863 


.09 137 


.10 960 


.20 097 


59 


2 


.79 918 


.89 030 


.90 889 


.09111 


.10970 


.20 082 


58 


3 


.79 934 


.89 020 


.90 914 


.09 086 


.10 980 


.20 066 


57 


4 


.79 950 


.89 009 


.90 940 


.09 060 


.10 991 


.20 050 


56 


5 


9.79 965 


9.88 999 


9.90 966 


0.09 034 


0.11 001 


0.20 035 


55 


6 


.79 981 


.88 989 


.90 992 


.09 008 


.11011 


.20 019 


54 


7 


.79 996 


.88 978 


.91 018 


.08 982 


.11 022 


.20 004 


53 


8 


.80 012 


.88968 


.91 043 


.08 957 


.11 032 


.19 988 


52 


9 


.80 027 


.88958 


.91 069 


.08 931 


.11 042 


.19 973 


51 


10 


9.80 043 


9.88 948 


9.91 095 


0.08 905 


0.11 052 


0.19 957 


50 


11 


.80058 


.88 937 


.91 121 


.08 879 


.11 063 


.19 942 


49 


12 


.80 074 


.88927 


.91 147 


.08 853 


.11 073 


.19 926 


48 


13 


.80 089 


.88 917 


.91 172 


.08 828 


.11083 


.19911 


47 


14 


.80 105 


.88906 


.91 198 


.08 802 


.11 094 


.19 895 


46 


15 


9.80 120 


9.88 896 


9.91 224 


0.08 776 


0.11 104 


0.19 880 


45 


16 


.80136 


.88886 


.91 250 


.08 750 


.11 114 


.19864 


44 


17 


.80 151 


.88 875 


.91 276 


.08 724 


.11 125 


.19 849 


43 


18 


.80 166 


.88865 


.91 301 


.08 699 


.11 135 


.19834 


42 


19 


.80 182 


.88855 


.91 327 


.08 673 


.11 145 


.19818 


41 


20 


9.80 197 


9.88 844 


9.91 353 


0.08 647 


0.11 156 


0.19 803 


40 


21 


.80 213 


.88 834 


.91 379 


.08 621 


.11 166 


.19 787 


39 


22 


.80 228 


.88824 


.91 404 


.08 596 


.11 176 


.19 772 


38 


23 


.80 244 


.88 813 


.91 430 


.08 570 


.11 187 


.19 756 


37 


24 


.80 259 


.88803 


.91 456 


.08 544 


.11 197 


.19 741 


36 


25 


9.80 274 


9.88 793 


9.91 482 


0.08 518 


0.11 207 


0.19 726 


35 


26 


.80 290 


.88 782 


.91 507 


.08 493 


.11 218 


.19710 


34 


27 


.80 305 


.88 772 


.91 533 


.08 467 


.11 228 


.19 695 


33 


28 


.80 320 


.88 761 


.91 559 


.08 441 


.11 239 


.19 680 


32 


29 


.80 336 


.88 751 


.91 585 


.08 415 


.11 249 


.19 664 


31 


30 


9.80 351 


9.88 741 


9.91 610 


0.08 390 


0.11 259 


0.19 649 


30 


31 


.80 366 


.88730 


.91 636 


.08364 


.11 270 


.19 634 


29 


32 


.80 382 


.88 720 


.91 662 


.08 338 


.11 280 


.19618 


28 


33 


.80 397 


.88 709 


.91 688 


.08 312 


.11291 


.19 603 


27 


34 


.80 412 


.88 699 


.91 713 


.08 287 


.11 301 


.19 588 


26 


35 


9.80 428 


9.88 688 


9.91 739 


0.08 261 


0.11312 


0.19 572 


25 


36 


.80 443 


.88 678 


.91 765 


.08 235 


.11 322 


.19 557 


24 


37 


.80 458 


.88 668 


.91 791 


.08 209 


.11332 


.19 542 


23 


38 


.80 473 


.88 657 


.91 816 


.08184 


.11343 


.19 527 


22 


39 


.80 489 


.88 647 


.91842 


.08 158 


.11 353 


.19511 


21 


40 


9.80 504 


9.88 636 


9.91 868 


0.08 132 


0.11 364 


0.19 496 


20 


41 


.80 519 


.88 626 


.91 893 


.08 107 


.11 374 


.19481 


19 


42 


.80 534 


.88615 


.91 919 


.08 081 


.11385 


.19466 


18 


43 


.80 550 


.88605 


.91 945 


.08 055 


.11 395 


.19450 


17 


44 


.80 565 


.88594 


.91 971 


.08 029 


.11406 


.19 435 


16 


45 


9.80 580 


9.88 584 


9.91 996 


0.08 004 


0.11416 


0.19 420 


15 


46 


.80 595 


.88 573 


.92 022 


.07 978 


.11427 


.19 405 


14 


47 


.80 610 


.88 563 


.92048 


.07 952 


.11437 


.19 390 


13 


48 


.80 625 


.88 552 


.92 073 


.07 927 


.11448 


.19 375 


12 


49 


.80641 


.88 542 


.92 099 


.07 901 


.11458 


.19 359 


11 


50 


9.80 656 


9.88 531 


9.92 125 


0.07 875 


0.11 469 


0.19344 


10 


51 


.80671 


.88 521 


.92 150 


.07 850 


.11479 


.19 329 


9 


52 


.80686 


.88 510 


.92 176 


.07 824 


.11 490 


.19314 


8 


53 


.80701 


.88 499 


.92 202 


.07 798 


.11 501 


.19 299 


7 


54 


.80716 


.88 489 


.92 227 


.07 773 


.11511 


.19284 


6 


55 


9.80 731 


9.88 478 


9.92 253 


0.07 747 


0.11 522 


0.19 269 


5 


56 


.80746 


.88468 


.92 279 


.07 721 


.11 532 


.19 254 


4 


57 


.80762 


.88 457 


.92304 


.07 696 


.11 543 


.19 238 


3 


58 


.80 777 


.88447 


.92 330 


.07 670 


.11 553 


.19 223 


2 


59 


.80 792 


.88 436 


.92 356 


.07 644 


.11564 


.19 208 


1 


60 


9. so s()7 


9.88 425 


9.92 381 


0.07 619 


0.11 575 


0.19 193 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec ' 



129 (309) 



(230) 50 



236 



Table 4. Trigonometric Logarithms 



40 (220) 



(319) 139 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







9.80 807 


9.88 425 


9.92 381 


0.07 619 


0.11575 


0.19 193 


60 


1 


.80 822 


.88415 


.92 407 


.07 593 


.11585 


.19 178 


59 


2 


.80 837 


.88 404 


.92 433 


.07 567 


.11 596 


.19 163 


58 


3 


.80 852 


.88 394 


.92 458 


.07 542 


.11 606 


.19 148 


57 


4 


.80 867 


.88383 


.92 484 


.07 516 


.11 617 


.19 133 


56 


5 


9.80 882 


9.88 372 


9.92 510 


0.07 490 


0.11 628 


0.19 118 


55 


6 


.80 897 


.88 362 


.92 535 


.07 465 


.11 638 


.19 103 


54 


7 


.80 912 


.88 351 


.92 561 


.07 439 


.11 649 


.19 088 


53 


8 


.80 927 


.88340 


.92 587 


.07 413 


.11 660 


.19 073 


52 


9 


.80 942 


.88 330 


.92 612 


.07 388 


.11 670 


.19 058 


51 


10 


9.80 957 


9.88 319 


9.92 638 


0.07 362 


0.11 681 


0.19 043 


50 


11 


.80 972 


.88308 


.92 663 


.07 337 


.11692 


.19 028 


49 


12 


.80 987 


.88 298 


.92 689 


.07311 


.11 702 


.19013 


48 


13 


.81 002 


.88 287 


.92 715 


.07 285 


.11713 


.18 998 


47 


14 


.81 017 


.88276 


.92 740 


07 260 


.11 724 


.18 983 


46 


15 


9.81 032 


9.88 266 


9.92 766 


0.07 234 


0.11 734 


0.18 968 


45 


16 


.81 047 


.88 255 


.92 792 


.07 208 


.11745 


.18 953 


44 


17 


.81 061 


.88 244 


.92 817 


.07 183 


.11 756 


.18 939 


43 


18 


.81076 


.88 234 


.92 843 


.07 157 


.11 766 


.18 924 


42 


19 


.81 091 


.88223 


.92 868 


07 132 


.11 777 


.18 909 


41 


20 


9.81 106 


9.88 212 


9.92 894 


0.07 106 


0.11 788 


0.18 894 


40 


21 


.81 121 


.88 201 


.92 920 


.07 080 


.11 799 


.18 879 


39 


22 


.81 136 


.88 191 


.92 945 


.07 055 


.11 809 


.18 864 


38 


23 


.81 151 


.88 180 


.92 971 


.07 029 


.11 820 


.18849 


37 


24 


.81 166 


.88 169 


.92996 


.07 004 


.11 831 


.18 834 


36 


25 


9.81 180 


9.88 158 


9.93 022 


0.06 978 


0.11 842 


0.18 820 


35 


26 


.81 195 


.88 148 


.93 048 


.06 952 


.11 852 


.18 805 


34 


27 


.81 210 


.88 137 


.93 073 


.06 927 


.11 863 


.18 790 


33 


28 


.81 225 


.88 126 


.93 099 


.06 901 


.11 874 


.18 775 


32 


29 


.81 240 


.88 115 


.93 124 


.06 876 


.11 885 


.18 760 


31 


30 


9.81 254 


9.88 105 


9.93 150 


0.06 850 


0.11 895 


0.18 746 


30 


31 


.81 269 


.88 094 


.93 175 


.06 825 


.11 906 


.18731 


29 


32 


.81 284 


.88 083 


.93 201 


.06 799 


.11 917 


.18716 


28 


33 


.81 299 


.88 072 


.93 227 


.06 773 


.11 928 


.18701 


27 


34 


.81 314 


.88 061 


.93 252 


.06 748 


.11 939 


.18 686 


26 


35 


9.81 328 


9.88 051 


9.93 278 


0.06 722 


0.11 949 


0.18 672 


25 


36 


.81 343 


.88040 


.93 303 


.06 697 


.11 960 


.18 657 


24 


37 


.81 358 


.88029 


.93 329 


.06 671 


.11971 


.18 642 


23 


38 


.81 372 


.88 018 


.93 354 


.06 646 


.11 982 


.18628 


22 


39 


.81 387 


.88 007 


.93 380 


.06 620 


.11 993 


.18613 


21 


40 


9.81 402 


9.87 996 


9.93 406 


0.06 594 


0.12 004 


0.18598 


20 


41 


.81 417 


.87 985 


.93 431 


.06 569 


.12015 


.18 583 


19 


42 


.81 431 


.87 975 


.93 457 


.06 543 


.12 025 


.18 569 


18 


43 


.81 446 


.87 964 


.92 482 


.06 518 


.12 036 


.18554 


17 


44 


.81 461 


.87 953 


.93 508 


.06 492 


.12 047 


.18539 


16 


45 


9.81 475 


9.87 942 


9.93 533 


0.06 467 


0.12 058 


0.18525 


15 


46 


.81 490 


.87 931 


.93 559 


.06 441 


.12 069 


.18510 


14 


47 


.81 505 


.87 920 


.93584 


.06 416 


.12 080 


.18 495 


13 


48 


.81 519 


.87 909 


.93 610 


.06 390 


.12 091 


.18 481 


12 


49 


.81 534 


.87 898 


.93 636 


.06 364 


.12 102 


.18 466 


11 


50 


9.81 549 


9.87 887 


9.93 661 


0.06 339 


0.12 113 


0.18451 


10 


51 


.81 563 


.87 877 


.93 687 


.06 313 


.12 123 


.18437 


9 


52 


.81 578 


.87 866 


.93 712 


.06 288 


.12 134 


.18422 


8 


53 


.81 592 


.87 855 


.93 738 


.06 262 


.12 145 


.18 408 


7 


54 


.81 607 


.87844 


.93 763 


.06 237 


.12 156 


.18393 


6 


55 


9.81 622 


9.87 833 


9.93 789 


0.06211 


0.12 167 


0.18 378 


5 


56 


.81 636 


.87 822 


.93 814 


.06 186 


.12 178 


.18 364 


4 


57 


.81 651 


.87811 


.93 840 


.06 160 


.12 189 


.18 349 


3 


58 


.81 665 


.87 800 


.93 865 


.06 135 


.12 200 


.18 335 


2 


59 


.81 680 


.87 789 


.93 891 


.06 109 


.12211 


.18 320 


1 


60 


9.81 694 


9.87 778 


9.93 916 


0.06 084 


0.12 222 


.18 306 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



130 (310) 



(229) 49 



Table 4. Trigonometric Logarithms 



237 



41 (221) 



(318) 138 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







9.81 694 


9.87 778 


9.93 916 


0.06 084 


0.12222 


0.18306 


60 


1 


.81 709 


.87 767 


.93 942 


.06 058 


.12 233 


.18291 


59 


2 


.81 723 


.87 756 


.93 967 


.06 033 


.12 244 


.18277 


58 


3 


.81 738 


.87 745 


.93 993 


.06 007 


.12255 


.18262 


57 


4 


.81 752 


.87 734 


.94 018 


.05 982 


.12 266 


.18248 


56 


5 


9.81 767 


9.87 723 


9.94 044 


0.05 956 


0.12 277 


0.18233 


55 


6 


.81 781 


.87 712 


.94 069 


.05 931 


.12 288 


.18219 


54 


7 


.81 796 


.87 701 


.94 095 


.05 905 


.12 299 


.18204 


53 


8 


.81 810 


.87 690 


.94 120 


.05880 


.12310 


.18 190 


52 


9 


.81 825 


.87 679 


.94 146 


.05 854 


.12 321 


.18 175 


51 


10 


9.81 839 


9.87 668 


9.94 171 


0.05 829 


0.12332 


0.18 161 


50 


11 


.81854 


.87 657 


.94 197 


.05 803 


.12 343 


.18 146 


49 


12 


.81 868 


.87646 


.94 222 


-.05 778 


.12 354 


.18 132 


48 


13 


.81 882 


.87 635 


.94248 


.05 752 


.12 365 


.18 118 


47 


14 


.81 897 


.87 624 


.94 273 


.05 727 


.12 376 


.18 103 


46 


15 


9.81 911 


9.87 613 


9.94 299 


0.05 701 


0.12 387 


0.18 089 


45 


16 


.81 926 


.87 601 


.94324 


.05 676 


.12 399 


.18074 


44 


17 


.81 940 


.87 590 


.94 350 


.05 650 


.12410 


.18 060 


43 


18 


.81 955 


.87 579 


.94 375 


.05 625 


.12421 


.18 045 


42 


19 


.81 969 


.87 568 


.94401 


.05 599 


.12432 


.18031 


41 


20 


9.81 983 


9.87 557 


9.94 426 


0.05 574 


0.12 443 


0.18017 


40 


21 


.81 998 


.87 546 


.94 452 


.05 548 


.12 454 


.18 002 


39 


22 


.82 012 


.87 535 


.94 477 


.05 523 


.12 465 


.17 988 


38 


23 


.82 026 


.87 524 


.94 503 


.05 497 


.12 476 


.17 974 


37 


24 


.82041 


.87 513 


.94 528 


.05 472 


.12487 


.17 959 


36 


25 


9.82 055 


9.87 501 


9.94 554 


0.05 446 


0.12 499 


0.17 945 


35 


26 


.82 069 


.87 490 


.94 579 


.05 421 


.12510 


.17 931 


34 


27 


.82084 


.87 479 


.94604 


.05 396 


.12521 


.17916 


33 


28 


.82 098 


.87 468 


.94 630 


.05 370 


.12 532 


.17 902 


32 


29 


.82112 


.87 457 


.94 655 


.05 345 


.12 543 


.17 888 


31 


30 


9.82 126 


9.87 446 


9.94 681 


0.05 319 


0.12 554 


0.17 874 


30 


31 


.82 141 


.87 434 


.94706 


.05 294 


.12 566 


.17859 


29 


32 


.82 155 


.87 423 


.94 732 


.05 268 


.12577 


.17845 


28 


33 


.82 169 


.87 412 


.94 757 


.05 243 


.12 588 


.17831 


27 


34 


.82184 


.87 401 


.94783 


05 217 


.12 599 


.17816 


26 


35 


9.82 198 


9.87 390 


9.94 808 


0.05 192 


0.12610 


0.17 802 


25 


36 


.82 212 


.87 378 


.94 834 


.05 166 


.12 622 


.17 788 


24 


37 


.82 226 


.87 367 


.94 859 


.05 141 


.12 633 


.17 774 


23 


38 


.82 240 


.87 356 


.94884 


.05 116 


.12644 


.17 760 


22 


39 


.82 255 


.87 345 


.94 910 


.05 090 


.12 655 


.17 745 


21 


40 


9.82 269 


9.87 334 


9.94 935 


0.05 065 


0.12 666 


0.17731 


20 


41 


.82 283 


.87 322 


.94 961 


.05 039 


.12 678 


.17717 


19 


42 


.82 297 


.87311 


.94 986 


.05 014 


.12 689 


.17 703 


18 


43 


.82311 


.87 300 


.95 012 


.04988 


.12 700 


.17 689 


17 


44 


.82 326 


.87 288 


.95 037 


.04963 


.12712 


.17 674 


16 


45 


9.82 340 


9.87 277 


9.95 062 


0.04 938 


0.12 723 


0.17 660 


15 


46 


.82 354 


.87 266 


.95 088 


.04 912 


.12 734 


.17646 


14 


47 


.82 368 


.87 255 


.95113 


.04887 


.12 745 


.17 632 


13 


48 


82 382 


.87 243 


.95 139 


.04861 


.12 757 


.17618 


12 


49 


.82 396 


.87 232 


.95 164 


.04836 


.12 768 


.17604 


11 


50 


9.82 410 


9.87 221 


9.95 190 


0.04 810 


0.12 779 


0.17590 


10 


51 


.82 424 


.87 209 


.95 215 


.04785 


.12 791 


.17 576 


9 


52 


.82 439 


.87 198 


.95 240 


.04760 


.12 802 


.17561 


8 


53 


.82 453 


.87 187 


.95 266 


.04734 


.12813 


.17 547 


7 


54 


.82 467 


.87 175 


.95 291 


.04709 


.12 825 


.17533 


6 


55 


9.82 481 


9.87 164 


9.95 317 


0.04 683 


0.12 836 


0.17519 


5 


56 


.82 495 


.87 153 


.95 342 


.04658 


.12847 


.17 505 


4 


57 


.82 509 


.87 141 


.95 368 


.04632 


.12 859 


.17491 


3 


58 


.82 523 


.87 130 


.95 393 


.04607 


.12 870 


.17 477 


2 


59 


.82 537 


.87119 


.95418 


.04582 


.12881 


.17 463 


1 


60 


9.82 551 


9.87 107 


9.95 444 


0.04 556 


0.12 893 


0.17 449 







Cos 


Sin 


Cot 


Tail 


Csc 


Sec 


' 



131 (311) 



(228) 48 



238 



Table 4. Trigonometric Logarithms 



42 (222) 



(317) 137 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







9.82 551 


9.87 107 


9.95 444 


0.04 556 


0.12 893 


0.17 449 


60 


1 


.82 565 


.87 096 


.95 469 


.04 531 


.12904 


.17 435 


59 


2 


.82 579 


.87 085 


.95 495 


.04 505 


.12915 


.17 421 


58 


3 


.82 593 


.87 073 


.95 520 


.04 480 


.12 927 


.17 407 


57 


4 


.82 607 


.87 062 


.95 545 


.04455 


.12 938 


.17 393 


56 


5 


9.82 621 


9.87 050 


9.95 571 


0.04 429 


0.12 950 


0.17 379 


55 


6 


.82 635 


.87 039 


.95 596 


.04404 


.12 961 


.17 365 


54 


7 


.82 649 


.87 028 


.95 622 


.04378 


.12 972 


.17 351 


53 


8 


.82 663 


.87 016 


.95 647 


.04353 


.12984 


.17 337 


52 


9 


.82 677 


.87 005 


.95 672 


.04 328 


.12 995 


.17 323 


51 


10 


9.82 691 


9.86 993 


9.95 698 


0.04 302 


0.13 007 


0.17 309 


50 


11 


.82 705 


.86 982 


.95 723 


.04 277 


.13018 


.17 295 


49 


12 


.82 719 


.86 970 


.95 748 


.04252 


.13 030 


.17 281 


48 


13 


.82 733 


.86 959 


.95 774 


.04 226 


.13 041 


.17 267 


47 


14 


.82 747 


.86 947 


.95 799 


.04201 


.13 053 


.17 253 


46 


15 


9.82 761 


9.86 936 


9.95 825 


0.04 175 


0.13 064 


0.17 239 


45 


16 


.82 775 


.86 924 


.95 850 


.04150 


.13 076 


.17 225 


44 


17 


.82 788 


.86 913 


.95 875 


.04125 


.13 087 


.17212 


43 


18 


.82 802 


.86 902 


.95 901 


.04 099 


.13 098 


.17 198 


42 


19 


.82 816 


.86 890 


.95 926 


.04074 


.13 110 


.17 184 


41 


20 


9.82 830 


9.86 879 


9.95 952 


0.04 048 


0.13 121 


0.17 170 


40 


21 


.82 844 


.86 867 


.95 977 


.04 023 


.13 133 


.17 156 


39 


22 


.82 858 


.86 855 


.96 002 


.03 998 


.13 145 


.17 142 


38 


23 


.82 872 


.86844 


.96 028 


.03972 


.13 156 


.17 128 


37 


24 


.82 885 


.86 832 


.96 053 


.03 947 


.13 168 


.17 115 


36 


25 


9.82 899 


9.86 821 


9.96 078 


0.03 922 


0.13 179 


0.17 101 


35 


26 


.82 913 


.86 809 


.96 104 


.03 896 


.13 191 


.17 087 


34 


27 


.82 927 


.86 798 


.96 129 


.03 871 


.13 202 


.17 073 


33 


28 


.82 941 


.86 786 


.96 155 


.03 845 


.13214 


.17 059 


32 


29 


.82 955 


.86 775 


.96 180 


.03 820 


.13 225 


.17 045 


31 


30 


9.82 968 


9.86 763 


9.96 205 


0.03 795 


0.13 237 


0.17 032 


30 


31 


.82 982 


.86 752 


.96 231 


.03 769 


.13 248 


.17018 


29 


32 


.82 996 


.86 740 


.96 256 


.03 744 


.13 260 


.17 004 


28 


33 


.83 010 


.86 728 


.96 281 


.03 719 


.13 272 


.16 990 


27 


34 


.83023 


.86 717 


.96 307 


.03 693 


.13283 


.16977 


26 


35 


9.83 037 


9.86 705 


9.96 332 


0.03 668 


0.13 295 


0.16 963 


25 


36 


.83 051 


.86 694 


.96 357 


.03 643 


.13 306 


.16 949 


24 


37 


.83065 


.86 682 


.96383 


.03 617 


.13318 


.16 935 


23 


38 


.83 078 


.86 670 


.96 408 


.03 592 


.13 330 


.16 922 


22 


39 


.83092 


.86 659 


.96 433 


.03 567 


.13 341 


.16 908 


21 


40 


9.83 106 


9.86 647 


9.96 459 


0.03 541 


0.13 353 


0.16 894 


20 


41 


.83 120 


.86 635 


.96 484 


.03 516 


.13 365 


.16 880 


19 


42 


.83 133 


.86 624 


.96 510 


.03 490 


.13 376 


.16 867 


18 


43 


.83 147 


.86 612 


.96 535 


.03 465 


.13 388 


.16 853 


17 


44 


.83 161 


.86 600 


.96 560 


.03 440 


.13 400 


.16 839 


16 


45 


9.83 174 


9.86 589 


9.96 586 


0.03 414 


0.13411 


0.16 826 


15 


46 


.83 188 


.86 577 


.96611 


.03 389 


.13 423 


.16812 


14 


47 


.83 202 


.86 565 


.96 636 


.03 364 


.13 435 


.16 798 


13 


48 


.83 215 


.86 554 


.96 662 


.03 338 


.13 446 


.16 785 


12 


49 


.83229 


.86 542 


.96 687 


.03 313 


.13 458 


.16771 


11 


50 


9.83 242 


9.86 530 


9.96 712 


0.03 288 


0.13 470 


0.16 758 


10 


51 


.83 256 


.86 518 


.96 738 


.03 262 


.13 482 


.16 744 


9 


52 


.83 270 


.86 507 


.96 763 


.03 237 


.13 493 


.16 730 


8 


53 


.83283 


.86 495 


.96 788 


.03 212 


.13 505 


.16717 


7 


54 


.83 297 


.86 483 


.96 814 


.03 186 


.13517 


.16 703 


6 


55 


9.83 310 


9.86 472 


9.96 839 


0.03 161 


0.13 528 


0.16 690 


5 


56 


.83 324 


.86 460 


.96864 


.03 136 


.13 540 


.16 676 


4 


57 


.83 338 


.86 448 


.96 890 


.03 110 


.13 552 


.16 662 


3 


58 


.83 351 


.86 436 


.96 915 


.03 085 


.13564 


.16649 


2 


59 


.83 365 


.86 425 


.96 940 


.03 060 


.13575 


.16 635 


1 


60 


9.83 378 


9.86 413 


9.96 966 


0.03 034 


0.13 587 


0.16 622 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



132 (312) 



(227) 47 



Table 4. Trigonometric Logarithms 



239 



43 (223) 



(316) 136 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







9.83 378 


9.86413 


9.96 966 


0.03 034 


0.13587 


0.16622 


60 


1 


.83 392 


.86 401 


.96 991 


.03 009 


.13 599 


.16 608 


59 


2 


.83405 


.86 389 


.97 016 


.02984 


.13611 


.16 595 


58 


3 


.83 419 


.86 377 


.97 042 


.02 958 


.13 623 


.16 581 


57 


4 


.83 432 


.86366 


.97 067 


.02 933 


.13 634 


.16 568 


56 


5 


9.83 446 


9.86 354 


9.97 092 


0.02 908 


0.13 646 


0.16 554 


55 


6 


.83459 


.86 342 


.97 118 


.02 882 


.13 658 


.16 541 


54 


7 


.83473 


.86 330 


.97 143 


.02 857 


.13 670 


.16 527 


53 


8 


.83486 


.86 318 


.97 168 


.02 832 


.13682 


.16 514 


52 


9 


.83500 


.86 306 


.97 193 


.02 807 


.13 694 


.16500 


51 


10 


9.83 513 


9.86 295 


9.97 219 


0.02 781 


0.13 705 


0.16487 


50 


11 


.83527 


.86283 


.97 244 


.02 756 


.13717 


.16473 


49 


12 


.83 540 


.86271 


.97 269 


.02 731 


.13 729 


.16 460 


48 


13 


.83554 


.86 259 


.97 295 


.02 705 


.13 741 


.16 446 


47 


14 


.83567 


.86 247 


.97 320 


.02 680 


.13 753 


.16433 


46 


15 


9.83 581 


9.86 235 


9.97 345 


0.02 655 


0.13 765 


0.16419 


45 


16 


.83594 


.86 223 


.97 371 


.02 629 


.13 777 


.16 406 


44 


17 


.83608 


.86211 


.97 396 


.02604 


.13 789 


.16 392 


43 


18 


.83621 


.86 200 


.97 421 


.02 579 


.13 800 


.16 379 


42 


19 


.83634 


.86 188 


.97 447 


.02 553 


.13812 


.16 366 


41 


20 


9.83 648 


9.86 176 


9.97 472 


0.02 528 


0.13 824 


0.16352 


40 


21 


.83661 


.86164 


.97 497 


.02 503 


.13 836 


.16 339 


39 


22 


.83674 


.86 152 


.97 523 


.02 477 


.13848 


.16326 


38 


23 


.83688 


.86 140 


.97 548 


.02 452 


.13 860 


.16312 


37 


24 


.83701 


.86 128 


.97 573 


.02 427 


.13 872 


.16 299 


36 


25 


9.83 715 


9.86 116 


9.97 598 


0.02 402 


0.13 884 


0.16285 


35 


26 


.83728 


.86104 


.97 624 


.02 376 


.13 896 


.16 272 


34 


27 


.83741 


.86 092 


.97 649 


.02 351 


.13 908 


.16 259 


33 


28 


.83755 


.86 080 


.97 674 


.02 326 


.13 920 


.16 245 


32 


29 


.83768 


.86 068 


.97 700 


.02 300 


.13 932 


.16232 


31 


30 


9.83 781 


9.86 056 


9.97 725 


0.02 275 


0.13 944 


0.16219 


30 


31 


.83 795 


.86044 


.97 750 


.02 250 


.13 956 


.16 205 


29 


32 


.83 808 


.86 032 


.97 776 


.02 224 


.13 968 


.16 192 


28 


33 


.83 821 


.86 020 


.97 801 


.02 199 


.13 980 


.16 179 


27 


34 


.83834 


.86 008 


.97 826 


.02 174 


.13 992 


.16 166 


26 


35 


9.83848 


9.85 996 


9.97 851 


0.02 149 


0.14 004 


0.16 152 


25 


36 


.83861 


.85984 


.97 877 


.02 123 


.14016 


.16 139 


24 


37 


.83874 


.85972 


.97 902 


.02 098 


.14 028 


.16 126 


23 


38 


.83887 


.85 960 


.97 927 


.02 073 


.14 040 


.16 113 


22 


39 


.83901 


.85948 


.97 953 


.02 047 


.14 052 


.16 099 


21 


40 


9.83 914 


9.85 936 


9.97 978 


0.02 022 


0.14 064 


0.16 086 


20 


41 


.83927 


.85924 


.98 003 


.01 997 


.14 076 


.16 073 


19 


42 


.83940 


.85912 


.98 029 


.01 971 


.14 088 


.16 060 


18 


43 


.83954 


.85900 


.98054 


.01 946 


.14 100 


.16 046 


17 


44 


.83967 


.85888 


.98 079 


.01 921 


.14 112 


.16 033 


16 


45 


9.83 980 


9.85 876 


9.98 104 


0.01 896 


0.14 124 


0.16020 


15 


46 


.83993 


.85864 


.98 130 


.01 870 


.14 136 


.16 007 


14 


47 


.84006 


.85851 


.98 155 


.01845 


.14 149 


.15 994 


13 


48 


.84020 


.85839 


.98 180 


.01 820 


.14 161 


.15 980 


12 


49 


.84033 


.85827 


.98 206 


.01 794 


.14 173 


.15 967 


11 


50 


9.84046 


9.85 815 


9.98 231 


0.01 769 


0.14 185 


0.15 954 


10 


51 


.84059 


.85 803 


.98 256 


.01 744 


.14 197 


.15 941 


9 


52 


.84072 


.85791 


.98 281 


.01 719 


.14 209 


.15 928 


8 


53 


.84085 


.85 779 


.98 307 


.01 693 


.14 221 


.15915 


7 


54 


.84098 


.85766 


.98 332 


.01 668 


.14 234 


.15 902 


6 


55 


9.84 112 


9.85 754 


9.98 357 


0.01 643 


0.14 246 


0.15 888 


5 


56 


.84 125 


.85742 


.98 383 


.01 617 


.14 258 


.15 875 


4 


57 


.84138 


.85730 


.98 408 


.01 592 


.14 270 


.15862 


3 


58 


.84151 


.85718 


.98 433 


.01 567 


.14 282 


.15849 


2 


59 


.84 164 


.85706 


.98 458 


.01 542 


.14 294 


.15836 


1 


60 


9.84 177 


9.85 693 


9.98 484 


0.01 516 


0.14 307 


0.15 823 







Cos 


Sin 


Cot 


Tan 


Csc 


S<-- 


' 



133 (313) 



(226) 46 



240 



Table 4. Trigonometric Logarithms 



44 (224) 



(315) 135 



' 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 







9.84 177 


9.85 693 


9.98 484 


0.01 516 


0.14 307 


0.15 823 


60 


1 


.84 190 


.85 681 


.98 509 


.01 491 


.14319 


.15810 


59 


2 


.84203 


.85 669 


.98 534 


.01 466 


.14 331 


.15 797 


58 


3 


.84216 


.85 657 


.98 560 


.01 440 


.14 343 


.15784 


57 


4 


.84229 


.85645 


.98 585 


.01 415 


.14 355 


.15 771 


56 


5 


9.84 242 


9.85 632 


9.98 610 


0.01 390 


0.14 368 


0.15 758 


55 


6 


.84 255 


.85 620 


.98 635 


.01 365 


.14 380 


.15 745 


54 


7 


.84269 


.85 608 


.98 661 


.01 339 


.14 392 


.15 731 


53 


8 


.84282 


.85 596 


.98 686 


.01 314 


.14 404 


.15718 


52 


9 


.84295 


.85 583 


.98 711 


.01 289 


.14417 


.15 705 


51 


10 


9.84 308 


9.85 571 


9.98 737 


0.01 263 


0.14 429 


0.15 692 


50 


11 


.84 321 


.85 559 


.98 762 


.01 238 


.14 441 


.15 679 


49 


12 


.84 334 


.85 547 


.98 787 


.01 213 


.14 453 


.15 666 


48 


13 


.84347 


.85 534 


.98 812 


.01 188 


.14 466 


.15 653 


47 


14 


.84360 


.85 522 


.98 838 


.01 162 


.14478 


.15 640 


46 


15 


9.84 373 


9.85 510 


9.98 863 


0.01 137 


0.14 490 


0.15 627 


45 


16 


.84 385 


.85 497 


.98 888 


.01 112 


.14 503 


.15615 


44 


17 


.84 398 


.85485 


.98 913 


.01 087 


.14515 


.15 602 


43 


18 


.84411 


.85 473 


.98 939 


.01 061 


.14 527 


.15 589 


42 


19 


.84424 


.85 460 


.98 964 


.01 036 


.14 540 


.15 576 


41 


20 


9.84 437 


9.85 448 


9.98 989 


0.01 Oil 


0.14 552 


0.15 563 


40 


21 


.84450 


.85 436 


.99 015 


.00 985 


.14 564 


.15 550 


39 


22 


.84463 


.85 423 


.99 040 


.00 960 


.14 577 


.15537 


38 


23 


.84476 


.85411 


.99 065 


.00 935 


.14 589 


.15 524 


37 


24 


.84 489 


.85 399 


.99 090 


.00 910 


.14 601 


.15511 


36 


25 


9.84 502 


9.85 386 


9.99 116 


0.00 884 


0.14 614 


0.15 498 


35 


26 


.84 515 


.85 374 


.99 141 


.00859 


.14 626 


.15485 


34 


27 


.84528 


.85 361 


.99 166 


.00 834 


.14 639 


.15 472 


33 


28 


.84540 


.85 349 


.99 191 


.00 809 


.14651 


.15 460 


32 


29 


.84553 


.85 337 


.99 217 


.00 783 


.14 663 


.15 447 


31 


30 


9.84 566 


9.85 324 


9.99 242 


0.00 758 


0.14 676 


0.15 434 


30 


31 


.84579 


.85 312 


.99 267 


.00 733 


.14 688 


.15421 


29 


32 


.84592 


.85 299 


.99 293 


.00 707 


.14 701 


.15 408 


28 


33 


.84 605 


.85 287 


.99 318 


.00 682 


.14713 


.15 395 


27 


34 


.84618 


.85274 


.99 343 


.00 657 


.14 726 


,15 382 


26 


35 


9.84 630 


9.85 262 


9.99 368 


0.00 632 


0.14 738 


0.15 370 


25 


36 


.84643 


.85 250 


.99 394 


.00 606 


.14 750 


.15 357 


24 


37 


.84656 


.85237 


.99 419 


.00 581 


.14 763 


.15 344 


23 


38 


.84669 


.85 225 


.99 444 


.00 556 


.14 775 


.15331 


22 


39 


.84682 


.85212 


.99 469 


.00 531 


.14788 


.15318 


21 


40 


9.84 694 


9.85 200 


9.99 495 


0.00 505 


0.14 800 


0.15 306 


20 


41 


.84707 


.85 187 


.99 520 


.00 480 


.14813 


.15 293 


19 


42 


.84720 


.85 175 


.99 545 


.00 455 


.14 825 


.15 280 


18 


43 


.84733 


.85 162 


.99 570 


.00 430 


.14 838 


.15 267 


17 


44 


.84745 


.85 150 


.99 596 


.00 404 


.14 850 


.15 255 


16 


45 


9.84 758 


9.85 137 


9.99 621 


0.00 379 


0.14 863 


0.15242 


15 


46 


.84771 


.85 125 


.99 646 


.00 354 


.14 875 


.15 229 


14 


47 


.84784 


.85 112 


.99 672 


.00 328 


.14 888 


.16216 


13 


48 


.84796 


.85 100 


.99 697 


.00303 


.14 900 


.15 204 


12 


49 


.84809 


.85 087 


.99 722 


.00 278 


.14 913 


.15 191 


11 


50 


9.84 822 


9.85 074 


9.99 747 


0.00 253 


0.14 926 


0.15 178 


10 


51 


.84835 


.85062 


.99 773 


.00 227 


.14 938 


.15 165 


9 


52 


.84847 


.85 049 


.99 798 


.00 202 


.14 951 


.15 153 


8 


53 


.84860 


.85 037 


.99 823 


.00 177 


.14 963 


.15 140 


7 


54 


.84873 


.85024 


.99848 


.00 152 


.14 976 


.15 127 


6 


55 


9.84 885 


9.85012 


9.99 874 


0.00 126 


0.14 988 


0.15 115 


5 


56 


.84898 


.84999 


.99 899 


.00 101 


.15 001 


.15 102 


4 


57 


.84911 


.84 986 


.99 924 


.00 076 


.15 014 


.15 089 


3 


58 


.84923 


.84 974 


.99 949 


.00 051 


.15 026 


.15 077 


2 


59 


.84936 


.84961 


.99 975 


.00 025 


.15 039 


.15 064 


1 


60 


9.84 949 


9.84 949 


0.00 000 


0.00 000 


0.15 051 


0.15051 







Cos 


Sin 


Cot 


Tan 


Csc 


Sec 


' 



134 (314) 



(225) 45 



Table 5. Meridional Parts 



241 



' 





1 


2 


3 


4 


5 


6 


7 


8 


9 


' 





0.0 


59.6 


119.2 


178.9 


238.6 


298.3 


358.2 


418.2 


478.3 


538.6 





1 


1.0 


60.6 


20.2 


79.9 


39.6 


99.3 


59.2 


19.2 


79.3 


39.6 


1 


2 


2.0 


61.6 


21.2 


80.8 


40.6 


300.3 


60.2 


20.2 


80.3 


40.6 


2 


3 


3.0 


62.6 


22.2 


81.8 


41.6 


01.3 


61.2 


21.2 


81.3 


41.6 


3 


4 


4.0 


63.6 


23.2 


82.8 


42.5 


02.3 


62.2 


22.2 


82.3 


42.6 


4 


5 


5.0 


64.6 


124.2 


183.8 


243.5 


303.3 


363.2 


423.2 


483.3 


543.6 


5 


6 


6.0 


65.6 


25.2 


84.8 


44.5 


04.3 


64.2 


24.2 


84.3 


44.6 


6 


7 


7.0 


66.5 


26.2 


85.8 


45.5 


05.3 


65.2 


25.2 


85.3 


45.6 


7 


8 


7.9 


67.5 


27.2 


86.8 


46.5 


06.3 


66.2 


26.2 


86.3 


46.6 


' 8 


9 


8.9 


68.5 


28.2 


87.8 


47.5 


07.3 


67.2 


27.2 


87.3 


47.6 


9 


10 


9.9 


69.5 


129.1 


188.8 


248.5 


308.3 


368.2 


428.2 


488.3 


548.6 


10 


11 


10.9 


70.5 


30.1 


89.8 


49.5 


09.3 


69.2 


29.2 


89.3 


49.6 


11 


12 


11.9 


71.5 


31.1 


90.8 


50.5 


10.3 


70.2 


30.2 


90.4 


50.6 


12 


13 


12.9 


72.5 


32.1 


91.8 


51.5 


11.3 


71.2 


31.2 


91.4 


51.7 


13 


14 


13.9 


73.5 


33.1 


92.8 


52.5 


12.3 


72.2 


32.2 


92.4 


52.7 


14 


15 


14.9 


74.5 


134.1 


193.8 


253.5 


313.3 


373.2 


433.2 


493.4 


553.7 


15 


16 


15.9 


75.5 


35.1 


94.8 


54.5 


14.3 


74.2 


34.2 


94.4 


54.7 


16 


17 


16.9 


76.5 


36.1 


95.8 


55.5 


15.3 


75.2 


35.2 


95.4 


55.7 


17 


18 


17.9 


77.5 


37.1 


96.8 


56.5 


16.3 


76.2 


36.2 


96.4 


56.7 


18 


19 


18.9 


78.5 


38.1 


97.8 


57.5 


17.3 


77.2 


37.2 


97.4 


57.7 


19 


20 


19.9 


79.5 


139.1 


198.8 


258.5 


318.3 


378.2 


438.2 


498.4 


558.7 


20 


21 


20.9 


80.5 


40.1 


99.7 


59.5 


19.3 


79.2 


39.2 


99.4 


59.7 


21 


22 


21.9 


81.5 


41.1 


200.7 


60.5 


20.3 


80.2 


40.2 


500.4 


60.7 


22 


23 


22.8 


82.4 


42.1 


01.7 


61.5 


21.3 


81.2 


41.2 


01.4 


61.7 


23 


24 


23.8 


83.4 


43.1 


02.7 


62.5 


22.3 


82.2 


42.2 


02.4 


62.7 


24 


25 


24.8 


84.4 


144.1 


203.7 


263.5 


323.3 


383.2 


443.2 


503.4 


563.7 


25 


26 


25.8 


85.4 


45.1 


04.7 


64.5 


24.3 


84.2 


44.2 


04.4 


64.7 


26 


27 


26.8 


86.4 


46.0 


05.7 


65.5 


25.3 


85.2 


45.2 


05.4 


65.7 


27 


28 


27.8 


87.4 


47.0 


06.7 


66.5 


26.3 


86.2 


46.2 


06.4 


66.8 


28 


29 


28.8 


88.4 


48.0 


07.7 


67.4 


27.3 


87.2 


47.2 


07.4 


67.8 


29 


30 


29.8 


89.4 


149.0 


208.7 


268.4 


328.3 


388.2 


448.2 


508.4 


568.8 


30 


31 


30.8 


90.4 


50.0 


09.7 


69.4 


29.3 


89.2 


49.2 


09.4 


69.8 


31 


32 


31.8 


91.4 


51.0 


10.7 


70.4 


30.3 


90.2 


50.2 


10.4 


70.8 


32 


33 


32.8 


92.4 


52.0 


11.7 


71.4 


31.3 


91.2 


51.2 


11.4 


71.8 


33 


34 


33.8 


93.4 


53.0 


12.7 


72.4 


32.3 


92.2 


52.2 


12.4 


72.8 


34 


35 


34.8 


94.4 


154.0 


213.7 


273.4 


333.3 


393.2 


453.2 


513.4 


573.8 


35 


36 


35.8 


95.4 


55.0 


14.7 


74.4 


34.3 


94.2 


54.3 


14.5 


74.8 


36 


37 


36.7 


96.4 


56.0 


15.7 


75.4 


35.3 


95.2 


55.3 


15.5 


75.8 


37 


38 


37.7 


97.3 


57.0 


16.7 


76.4 


36.2 


96.2 


56.3 


16.5 


76.8 


38 


39 


38.7 


98.3 


58.0 


17.7 


77.4 


37.2 


97.2 


57.3 


17.5 


77.8 


39 


40 


39.7 


99.3 


159.0 


218.7 


278.4 


338.2 


398.2 


458.3 


518.5 


578.8 


40 


41 


40.7 


100.3 


60.0 


19.7 


79.4 


39.2 


99.2 


59.3 


19.5 


79.9 


41 


42 


41.7 


01.3 


61.0 


20.6 


80.4 


40.2 


400.2 


60.3 


20.5 


80.9 


42 


43 


42.7 


02.3 


62.0 


21.6 


81.4 


41.2 


01.2 


61.3 


21.5 


81.9 


43 


44 


43.7 


03.3 


63.0 


22.6 


82.4 


42.2 


02.2 


62.3 


22.5 


82.9 


44 


45 


44.7 


104.3 


164.0 


223.6 


283.4 


343.2 


403.2 


463.3 


523.5 


583.9 


45 


46 


45.7 


05.3 


65.0 


24.6 


84.4 


44.2 


04.2 


64.3 


24.5 


84.9 


46 


47 


46.7 


06.3 


66.0 


25.6 


85.4 


45.2 


05.2 


65.3 


25.5 


85.9 


47 


48 


47.7 


07.3 


67.0 


26.6 


86.4 


46.2 


06.2 


66.3 


26.5 


86.9 


48 


49 


48.7 


08.3 


68.0 


27.6 


87.4 


47.2 


07.2 


67.3 


27.5 


87.9 


49 


50 


49.7 


109.3 


168.9 


228.6 


288.4 


348.2 


408.2 


468.3 


528.5 


588.9 


50 


51 


50.7 


10.3 


69.9 


29.6 


89.4 


49.2 


09.2 


69.3 


29.5 


89.9 


51 


52 


51.6 


11.3 


70.9 


30.6 


90.4 


50.2 


10.2 


70.3 


30.5 


90.9 


52 


53 


52.6 


12.3 


71.9 


31.6 


91.4 


51.2 


11.2 


71.3 


31.5 


91.9 


53 


54 


53.6 


13.2 


72.9 


32.6 


92.4 


52.2 


12.2 


72.3 


32.5 


93.0 


54 


55 


54.6 


114.2 


173.9 


233.6 


293.4 


353.2 


413.2 


473.3 


533.5 


594.0 


55 


56 


55.6 


15.2 


74.9 


34.6 


94.4 


54.2 


14.2 


74.3 


34.6 


95.0 


56 


57 


56.6 


16.2 


75.9 


35.6 


95.4 


55.2 


15.2 


75.3 


35.6 


96.0 


57 


58 


57.6 


17.2 


76.9 


36.6 


96.3 


56.2 


16.2 


76.3 


36.6 


97.0 


58 


59 


58.6 


18.2 


77.9 


37.6 


97.3 


57.2 


17.2 


77.3 


37.6 


98.0 


59 


60 


59.6 


119.2 


178.9 


238.6 


298.3 


358.2 


418.2 


478.3 


538.6 


599.0 


60 


' 





1 


2 


3 


4 


5 


6 


7 


8 


9 


/ 



242 



Table 5. Meridional Parts 



/ 


10 


11 


12 


13 


14 


15 


16 


17 


18 


19 


' 





599.0 


659.6 


720.5 


781.5 


842.8 


904.4 


966.3 


1028.5 


1091.0 


1153.9 





1 


600.0 


60.6 


21.5 


82.5 


43.9 


05.4 


67.3 


29.5 


92.0 


54.9 


1 


2 


01.0 


61.7 


22.5 


83.6 


44.9 


06.5 


68.3 


30.5 


93.1 


56.0 


2 


3 


02.0 


62.7 


23.5 


84.6 


45.9 


07.5 


69.4 


31.6 


94.1 


57.0 


3 


4 


03.0 


63.7 


24.5 


85.6 


46.9 


08.5 


70.4 


32.6 


95.2 


58.1 


4 


5 


604.1 


664.7 


725.5 


786.6 


847.9 


909.6 


971.4 


1033.7 


1096.2 


1159.1 


5 


6 


05.1 


65.7 


26.6 


87.6 


49.0 


10.6 


72.5 


34.7 


97.3 


60.2 


6 


7 


06.1 


66.7 


27.6 


88.7 


50.0 


11.6 


73.5 


35.7 


98.3 


61.2 


7 


8 


07.1 


67.7 


28.6 


89.7 


51.0 


12.6 


74.6 


36.8 


99.4 


62.3 


8 


9 


08.1 


68.7 


29.6 


90.7 


52.0 


13.7 


75.6 


37.8 


1100.4 


63.3 


9 


10 


609.1 


669.8 


730.6 


791.7 


853.1 


914.7 


976.6 


1038.9 


1101.4 


1164.4 


10 


11 


10.1 


70.8 


31.6 


92.7 


54.1 


15.7 


77.7 


39.9 


02.5 


65.4 


11 


12 


11.1 


71.8 


32.7 


93.8 


55.1 


16.8 


78.7 


40.9 


03.5 


66.5 


12 


13 


12.1 


72.8 


33.7 


94.8 


56.1 


17.8 


79.7 


42.0 


04.6 


67.5 


13 


14 


13.1 


73.8 


34.7 


95.8 


57.2 


18.8 


80.8 


43.0 


05.6 


68.6 


14 


15 


614.1 


674.8 


735.7 


796.8 


858.2 


919.8 


981.8 


1044.1 


1106.7 


1169.7 


15 


16 


15.2 


75.8 


36.7 


97.8 


59.2 


20.9 


82.8 


45.1 


07.7 


70.7 


16 


17 


16.2 


76.8 


37.7 


98.9 


60.2 


21.9 


83.9 


46.1 


08.8 


71.8 


17 


18 


17.2 


77.9 


38.8 


99.9 


61.3 


22.9 


84.9 


47.2 


09.8 


72.8 


18 


19 


18.2 


78.9 


39.8 


800.9 


62.3 


24.0 


85.9 


48.2 


10.9 


73.9 


19 


20 


619.2 


679.9 


740.8 


801.9 


863.3 


925.0 


987.0 


1049.3 


1111.9 


1174.9 


20 


21 


20.2 


80.9 


41.8 


02.9 


64.3 


26.0 


88.0 


50.3 


13.0 


76.0 


21 


22 


21.2 


81.9 


42.8 


04.0 


65.4 


27.1 


89.0 


51.3 


14.0 


77.0 


22 - 


23 


22.2 


82.9 


43.8 


05.0 


66.4 


28.1 


90.1 


52.4 


15.0 


78.1 


23 


24 


23.2 


83.9 


44.9 


06.0 


67.4 


29.1 


91.1 


53.4 


16.1 


79.1 


24 


25 


624.2 


684.9 


745.9 


807.0 


868.5 


930.1 


992.1 


1054.5 


1117.1 


1180.2 


25 


26 


25.3 


86.0 


46.9 


08.1 


69.5 


31.2 


93.2 


55.5 


18.2 


81.2 


26 


27 


26.3 


87.0 


47.9 


09.1 


70.5 


32.2 


94.2 


56.6 


19.2 


82.3 


27 


28 


27.3 


88.0 


48.9 


10.1 


71.5 


33.2 


95.3 


57.6 


20.3 


83.3 


28 


29 


28.3 


89.0 


49.9 


11.1 


72.6 


34.3- 


96.3 


58.6 


21.3 


84.4 


29 


30 


629.3 


690.0 


751.0 


812.1 


873.6 


935.3 


997.3 


1059.7 


1122.4 


1185.5 


30 


31 


30.3 


91.0 


52.0 


13.2 


74.6 


36.3 


98.4 


60.7 


23.4 


86.5 


31 


32 


31.3 


92.0 


53.0 


14.2 


75.6 


37.4 


99.4 


61.8 


24.5 


87.6 


32 


33 


32.3 


93.1 


54.0 


15.2 


76.7 


38.4 


1000.4 


62.8 


25.5 


88.6 


33 


34 


33.3 


94.1 


55.0 


16.2 


77.7 


39.4 


01.5 


63.9 


26.6 


89.7 


34 


35 


634.3 


695.1 


756.0 


817.3 


878.7 


940.5 


1002.5 


1064.9 


1127.6 


1190.7 


35 


36 


35.4 


96.1 


57.1 


18.3 


79.7 


41.5 


03.6 


65.9 


28.7 


91.8 


36 


37 


36.4 


97.1 


58.1 


19.3 


80.8 


42.5 


04.6 


67.0 


29.7 


92.8 


37 


38 


37.4 


98.1 


59.1 


20.3 


81.8 


43.6 


05.6 


68.0 


30.8 


93.9 


38 


39 


38.4 


99.1 


60.1 


21.3 


82.8 


44.6 


06.7 


69.1 


31.8 


95.0 


39 


40 


639.4 


700.2 


761.1 


822.4 


883.8 


945.6 


1007.7 


1070.1 


1132.9 


1196.0 


40 


41 


40.4 


01.2 


62.2 


23.4 


84.9 


46.7 


08.7 


71.2 


33.9 


97.1 


41 


42 


41.4 


02.2 


63.2 


24.4 


85.9 


47.7 


09.8 


72.2 


35.0 


98.1 


42 


43 


42.4 


03.2 


64.2 


25.4 


86.9 


48.7 


10.8 


73.2 


36.0 


99.2 


43 


44 


43.4 


04.2 


65.2 


26.5 


88.0 


49.7 


11.8 


74.3 


37.1 


1200.2 


44 


45 


644.5 


705.2 


766.2 


827.5 


889.0 


950.8 


1012.9 


1075.3 


1138.1 


1201.3 


45 


46 


45.5 


06.2 


67.3 


28.5 


90.0 


51.8 


13.9 


76.4 


39.2 


02.3 


46 


47 


46.5- 


07.3 


68.3 


29.5 


91.0 


52.8 


15.0 


77.4 


40.2 


03.4 


47 


48 


47.5 


08.3 


69.3 


30.5 


92.1 


53.9 


16.0 


78.5 


41.8 


04.5 


48 


49 


48.5 


09.3 


70.3 


31.6 


93.1 


54.9 


17.0 


79.5 


42.3 


05.5 


49 


50 


649.5 


710.3 


771.3 


832.6 


894.1 


955.9 


1018.1 


1080.5 


1143.4 


1206.6 


50 


51 


50.5 


11.3 


72.3 


33.6 


95.2 


57.0 


19.1 


81.6 


44.4 


07.6 


51 


52 


51.5 


12.3 


73.4 


34.6 


96.2 


58.0 


20.2 


82.6 


45.5 


08.7 


52 


53 


52.5 


13.4 


74.4 


35.7 


97.2 


59.0 


21.2 


83.7 


46.5 


09.7 


53 


54 


53.6 


14.4 


75.4 


36.7 


98.2 


60.1 


22.2 


84.7 


47.6 


10.8 


54 


55 


654.6 


715.4 


776.4 


837.7 


899.3 


961.1 


1023.3 


1085.8 


1148.6 


1211.8 


55 


56 


55.6 


16.4 


77.4 


38.7 


900.3 


62.1 


24.3 


86.8 


49.7 


12.9 


56 


57 


56.6 


17.4 


78.5 


39.8 


01.3 


63.2 


25.3 


87.9 


50.7 


14.0 


57 


58 


57.6 


18.4 


79.5 


40.8 


02.3 


64.2 


26.4 


88.9 


51.8 


15.0 


58 ' 


59 


58.6 


19.4 


80.5 


41.8 


03.4 


65.2 


27.4 


89.9 


52.8 


16.1 


59 


60 


659.6 


720.5 


781.5 


842.8 


904.4 


966.3 


1028.5 


1091.0 


1153.9 


1217.1 


60 


' 


10 


11 


12 


13 


14 


15 


16 


17 


18 


19 


' 



Table 5. Meridional Parts 



243 



' 


20 


21 


22 


23 


24 


25 


26 


27 


28 


29 


' 





1217.1 


1280.8 


1344.9 


1409.5 


1474.5 


1540.1 


1606.2 


1672.9 


1740.2 


1808.1 





1 


18.2 


81.9 


46.0 


10.6 


75.6 


41.2 


07.3 


74.0 


41.3 


09.2 


1 


2 


19.3 


82.9 


47.1 


11.6 


76.7 


42.3 


08.4 


75.1 


42.4 


10.4 


2 


3 


20.3 


84.0 


48.1 


12.7 


77.8 


43.4 


09.5 


76.2 


43.6 


11.5 


3 


4 


21.4 


85.1 


49.2 


13.8 


78.9 


44.5 


10.6 


77.4 


44.7 


12.6 


4 


5 


1222.4 


1286.1 


1350.3 


1414.9 


1480.0 


1545.6 


1611.7 


1678.5 


1745.8 


1813.8 


5 


6 


23.5 


87.2 


51.4 


16.0 


81.1 


46.7 


12.9 


79.6 


46.9 


14.9 


6 


7 


24.5 


88.3 


52.4 


17.1 


82.2 


47.8 


14.0 


80.7 


48.1 


16.1 


7 


8 


25.6 


89.3 


53.5 


18.1 


83.3 


48.9 


15.1 


81.8 


49.2 


17.2 


8 


9 


26.7 


90.4 


54.6 


19.2 


84.3 


50.0 


16.2 


82.9 


50.3 


18.3 


9 


10 


1227.7 


1291.5 


1355.7 


1420.3 


1485.4 


1551.1 


1617.3 


1684.1 


1751.5 


1819.5 


10 


11 


28.8 


92.5 


56.7 


21.4 


86.5 


52.2 


18.4 


85.2 


52.6 


20.6 


11 


12 


29.8 


93.6 


57.8 


22.5 


87.6 


53.3 


19.5 


86.3 


53.7 


21.8 


12 


13 


30.9 


94.7 


58.9 


23.5 


88.7 


54.4 


20.6 


87.4 


54.8 


22.9 


13 


14 


32.0 


95.7 


59.9 


24.6 


89.8 


55.5 


21.7 


88.5 


56.0 


24.0 


14 


15 


1233.0 


1296.8 


1361.0 


1425.7 


1490.9 


1556.6 


1622.8 


1689.7 


1757.1 


1825.2 


15 


16 


34.1 


97.9 


62.1 


26.8 


92.0 


57.7 


23.9 


90.8 


58.2 


26.3 


16 


17 


35.1 


98.9 


63.2 


27.9 


93.1 


58.8 


25.0 


91.9 


59.4 


27.5 


17 


18 


36.2 


1300.0 


64.2 


29.0 


94.2 


59.9 


26.2 


93.0 


60.5 


28.6 


18 


19 


37.3 


01.1 


65.3 


30.0 


95.2 


61.0 


27.3 


94.1 


61.6 


29.7 


19 


20 


1238.3 


1302.1 


1366.4 


1431.1 


1496.3 


1562.1 


1628.4 


1695.3 


1762.7 


1830.9 


20 


21 


39.4 


03.2 


67.5 


32.2 


97.4 


63.2 


29.5 


96.4 


63.9 


32.0 


21 


22 


40.4 


04.3 


68.5 


33.3 


98.5 


64.3 


30.6 


97.5 


65.0 


33.2 


22 


23 


41.5 


05.3 


69.6 


34.4 


99.6 


65.4 


31.7 


98.6 


66.1 


34.3 


23 


24 


42.6 


06.4 


70.7 


35.4 


1500.7 


66.5 


32.8 


99.7 


67.3 


35.4 


24 


25 


1243.6 


1307.5 


1371.8 


1436.5 


1501.8 


1567.6 


1633.9 


1700.9 


1768.4 


1836.6 


25 


26 


44.7 


08.5 


72.8 


37.6 


02.9 


68.7 


35.0 


02.0 


69.5 


37.7 


26 


27 


45.7 


09.6 


73.9 


38.7 


04.0 


69.8 


36.1 


03.1 


70.7 


38.9 


27 


28 


46.8 


10.7 


75.0 


39.8 


05.1 


70.9 


37.3 


04.2 


71.8 


40.0 


28 


29 


47.9 


11.7 


76.1 


40.9 


06.2 


72.0 


38.4 


05.3 


72.9 


41.2 


29 


30 


1248.9 


1312.8 


1377.1 


1442.0 


1507.3 


1573.1 


1639.5 


1706.5 


1774.1 


1842.3 


30 


31 


50.0 


13.9 


78.2 


43.0 


08.4 


74.2 


40.6 


07.6 


75.2 


43.4 


31 


32 


51.0 


14.9 


79.3 


44.1 


09.4 


75.3 


41.7 


08.7 


76.3 


44.6 


32 


33 


52.1 


16.0 


80.4 


45.2 


10.5 


76.4 


42.8 


09.8 


77.4 


45.7 


33 


34 


53.2 


17.1 


81.5 


46.3 


11.6 


77.5 


43.9 


10.9 


78.6 


46.9 


34 


35 


1254.2 


1318.2 


1382.5 


1447.4 


1512.7 


1578.6 


1645.0 


1712.1 


1779.7 


1848.0 


35 


36 


55.3 


19.2 


83.6 


48.5 


13.8 


79.7 


46.2 


13.2 


80.8 


49.2 


36 


37 


56.4 


20.3 


84.7 


49.5 


14.9 


80.8 


47.3 


14.3 


82.0 


50.3 


37 


38 


57.4 


21.4 


85.8 


50.6 


16.0 


81.9 


48.4 


15.4 


83.1 


51.4 


38 


39 


58.5 


22.4 


86.8 


51.7 


17.1 


83.0 


49.5 


16.6 


84.2 


52.6 


39 


40 


1259.5 


1323.5 


1387.9 


1452.8 


1518.2 


1584.1 


1650.6 


1717.7 


1785.4 


1853.7 


40 


41 


60.6 


24.6 


89.0 


53.9 


19.3 


85.2 


51.7 


18.8 


86.5 


54.9 


41 


42 


61.7 


25.6 


90.1 


55.0 


20.4 


86.3 


52.8 


19.9 


87.6 


56.0 


42 


43 


62.7 


26.7 


91.1 


56.1 


21.5 


87.4 


53.9 


21.1 


88.8 


57.2 


43 


44 


63.8 


27.8 


92.2 


57.1 


22.6 


88.5 


55.1 


22.2 


89.9 


58.3 


44 


45 


1264.9 


1328.9 


1393.3 


1458.2 


1523.7 


1589.6 


1656.2 


1723.3 


1791.1 


1859.5 


45 


46 


65.9 


29.9 


94.4 


59.3 


24.8 


90.7 


57.3 


24.4 


92.2 


60.6 


46 


47 


67.0 


31.0 


95.5 


60.4 


25.9 


91.8 


58.4 


25.5 


93.3 


61.8 


47 


48 


68.0 


32.1 


96.5 


61.5 


27.0 


92.9 


59.5 


26.7 


94.5 


62.9 


48 


49 


69.1 


33.1 


97.6 


62.6 


28.0 


94.1 


60.6 


27.8 


95.6 


64.0 


49 


50 


1270.2 


1334.2 


1398.7 


1463.7 


1529.1 


1595.2 


1661.7 


1728.9 


1796.7 


1865.2 


50 


51 


71.2 


35.3 


99.8 


64.8 


30.2 


96.3 


62.9 


30.0 


97.9 


66.3 


51 


52 


72.3 


36.3 


1400.9 


65.8 


31.3 


97.4 


64.0 


31.2 


99.0 


67.5 


52 


53 


73.4 


37.4 


01.9 


66.9 


32.4 


98.5 


65.1 


32.3 


1800.1 


68.6 


53 


54 


74.4 


38.5 


03.0 


68.0 


33.5 


99.6 


66.2 


33.4 


01.3 


69.8 


54 


55 


1275.5 


1339.6 


1404.1 


1469.1 


1534.6 


1600.7 


1667.3 


1734.5 


1802.4 


1870.9 


55 


56 


76.6 


40.6 


05.2 


70.2 


35.7 


01.8 


68.4 


35.7 


03.5 


72.1 


56 


57 


77.6 


41.7 


06.2 


71.3 


36.8 


02.9 


69.5 


36.8 


04.7 


73.2 


57 


58 


78.7 


42.8 


07.3 


72.4 


37.9 


04.0 


70.7 


37.9 


05.8 


74.4 


58 


59 


79.7 


43.8 


08.4 


73.5 


39.0 


05.1 


71.8 


39.1 


07.0 


75.5 


59 


60 


1280.8 


1344.9 


1409.5 


1474.5 


1540.1 


1606.2 


1672.9 


1740.2 


1808.1 


1876.7 


60 


' 


20 


21 


22 


23 


24 


25 


26 


27 


28 


29 


' 



244 



Table 5. Meridional Parts 



' 


30 


31 


32 


33 


34 


35 


36 


37 


38 


39 


' 





1876.7 


1946.0 


2016.0 


2086.8 


2158.4 


2230.9 


2304.2 


2378.5 


2453.8 


2530.2 





1 


77.8 


47.1 


17.2 


88.0 


59.6 


32.1 


05.5 


79.8 


55.1 


31.5 


1 


2 


79.0 


48.3 


18.3 


89.2 


60.8 


33.3 


06.7 


81.0 


56.4 


32.8 


2 


3 


80.1 


49.4 


19.5 


90.3 


62.0 


34.5 


07.9 


82.3 


57.6 


34.0 


3 


4 


81.3 


50.6 


20.7 


91.5 


63.2 


35.7 


09.2 


83.5 


58.9 


35.3 


4 


5 


1882.4 


1951.8 


2021.9 


2092.7 


2164.4 


2236.9 


2310.4 


2384.8 


2460.2 


2536.6 


5 


6 


83.6 


52.9 


23.0 


93.9 


65.6 


38.2 


11.6 


86.0 


61.4 


37.9 


6 


7 


84.7 


54.1 


24.2 


95.1 


66.8 


39.4 


12.9 


87.3 


62.7 


39.2 


7 


8 


85.9 


55.3 


25.4 


96.3 


68.0 


40.6 


14.1 


88.5 


64.0 


40.5 


8 


9 


87.0 


56.4 


26.6 


97.5 


69.2 


41.8 


15.3 


89.8 


65.2 


41.7 


9 


10 


1888.2 


1957.6 


2027.7 


2098.7 


2170.4 


2243.0 


2316.5 


2391.0 


2466.5 


2543.0 


10 


11 


89.3 


58.7 


28.9 


99.8 


71.6 


44.2 


17.8 


92.3 


67.8 


44.3 


11 


12 


90.5 


59.9 


30.1 


2101.0 


72.8 


45.5 


19.0 


93.5 


69.0 


45.6 


12 


13 


91.6 


61.1 


31.3 


02.2 


74.0 


46.7 


20.3 


94.8 


70.3 


46.9 


13 


14 


92.8 


62.2 


32.4 


03.4 


75.2 


47.9 


21.5 


96.0 


71.6 


48.2 


14 


15 


1893.9 


1963.4 


2033.6 


2104.6 


2176.4 


2249.1 


2322.7 


2397.3 


2472.8 


2549.5 


15 


16 


95.1 


64.6 


34.8 


05.8 


77.6 


50.3 


24.0 


98.5 


74.1 


50.7 


16 


17 


96.2 


65.7 


36.0 


07.0 


78.8 


51.6 


25.2 


99.8 


75.4 


52.0 


17 


18 


97.4 


66.9 


37.1 


08.2 


80.0 


52.8 


26.4 


2401.0 


76.6 


53.3 


18 


19 


98.5 


68.1 


38.3 


09.4 


81.2 


54.0 


27.7 


02.3 


77.9 


54.6 


19 


20 


1899.7 


1969.2 


2039.5 


2110.6 


2182.5 


2255.2 


2328.9 


2403.5 


2479.2 


2555.9 


20 


21 


1900.8 


70.4 


40.7 


11.8 


83.7 


56.4 


30.1 


04.8 


80.4 


57.2 


21 


22 


02.0 


71.5 


41.8 


12.9 


84.9 


57.7 


31.4 


06.0 


81.7 


58.5 


22 


23 


03.1 


72.7 


43.0 


14.1 


86.1 


58.9 


32.6 


07.3 


83.0 


59.8 


23 


24 


04.3 


73.9 


44.2 


15.3 


87.3 


60.1 


33.8 


08.5 


84.3 


61.0 


24 


25 


1905.5 


1975.0 


2045.4 


2116.5 


2188.5 


2261.3 


2335.1 


2409.8 


2485.5 


2562.3 


25 


26 


06.6 


76.2 


46.6 


17.7 


89.7 


62.5 


36.3 


11.1 


86.8 


63.6 


26 


27 


07.8 


77.4 


47.7 


18.9 


90.9 


63.8 


37.6 


12.3 


88.1 


64.9 


27 


28 


08.9 


78.5 


48.9 


20.1 


92.1 


65.0 


38.8 


13.6 


89.3 


66.2 


28 


29 


10.1 


79.7 


50.1 


21.3 


93.3 


66.2 


40.0 


14.8 


90.6 


67.5 


29 


30 


1911.2 


1980.9 


2051.3 


2122.5 


2194.5 


2267.4 


2341.3 


2416.1 


2491.9 


2568.8 


30 


31 


12.4 


82.0 


52.5 


23.7 


95.7 


68.7 


42.5 


17.3 


93.2 


70.1 


31 


32 


13.5 


83.2 


53.6 


24.9 


96.9 


69.9 


43.7 


18.6 


94.4 


71.4 


32 


33 


14.7 


84.4 


54.8 


26.1 


98.1 


71.1 


45.0 


19.8 


95.7 


72.7 


33 


34 


15.8 


85.5 


56.0 


27.3 


99.4 


72.3 


46.2 


21.1 


97.0 


73.9 


34 


35 


1917.0 


1986.7 


2057.2 


2128.5 


2200.6 


2273.5 


2347.5 


2422.3 


2498.3 


2575.2 


35 


36 


18.2 


87.9 


58.4 


29.6 


01.8 


74.8 


48.7 


23.6 


99.5 


76.5 


36 


37 


19.3 


89.1 


59.5 


30.8 


03.0 


76.0 


49.9 


24.9 


2500.8 


77.8 


37 


38 


20.5 


90.2 


60.7 


32.0 


04.2 


77.2 


51.2 


26.1 


02.1 


79.1 


38 


39 


21.6 


91.4 


61.9 


33.2 


05.4 


78.4 


52.4 


27.4 


03.4 


80.4 


39 


40 


1922.8 


1992.6 


2063.1 


2134.4 


2206.6 


2279.7 


2353.7 


2428.6 


2504.6 


2581.7 


40 


41 


23.9 


93.7 


64.3 


35.6 


07.8 


80.9 


54.9 


29.9 


05.9 


83.0 


41 


42 


25.1 


94.9 


65.5 


36.8 


09.0 


82.1 


56.1 


31.2 


07.2 


84.3 


42 


43 


26.3 


96.1 


66.6 


38.0 


10.2 


83.3 


57.4 


32.4 


08.5 


85.6 


43 


44 


27.4 


97.2 


67.8 


39.2 


11.5 


84.6 


58.6 


33.7 


09.7 


86.9 


44 


45 


1928.6 


1998.4 


2069.0 


2140.4 


2212.7 


2285.8 


2359.9 


2434.9 


2511.0 


2588.2 


45 


46 


29.7 


99.6 


70.2 


41.6 


13.9 


87.0 


61.1 


36.2 


12.3 


89.5 


46 


47 


30.9 


2000.7 


71.4 


42.8 


15.1 


88.3 


62.4 


37.4 


13.6 


90.8 


47 


48 


32.0 


01.9 


72.6 


44.0 


16.3 


89.5 


63.6 


38.7 


14.8 


92.1 


48 


49 


33.2 


03.1 


73.7 


45.2 


17.5 


90.7 


64.8 


40.0 


16.1 


93.4 


49 


50 


1934.4 


2004.3 


2074.9 


2146.4 


2218.7 


2291.9 


2366.1 


2441.2 


2517.4 


2594.7 


50 


51 


35.5 


05.4 


76.1 


47.6 


19.9 


93.2 


67.3 


42.5 


18.7 


96.0 


51 


52 


36.7 


06.6 


77.3 


48.8 


21.1 


94.4 


68.6 


43.7 


20.0 


97.3 


52 


53 


37.8 


07.8 


78.5 


50.0 


22.4 


95.6 


69.8 


45.0 


21.2 


98.5 


53 


54 


39.0 


08.9 


79.7 


51.2 


23.6 


96.9 


71.1 


46.3 


22.5 


99.8 


54 


55 


1940.2 


2010.1 


2080.8 


2152.4 


2224.8 


2298.1 


2372.3 


2447.5 


2523.8 


2601.1 


55 


56 


41.3 


11.3 


82.0 


53.6 


26.0 


99.3 


73.6 


48.8 


25.1 


02.4 


56 


57 


42.5 


12.5 


83.2 


54.8 


27.2 


2300.5 


74.8 


50.1 


26.4 


03.7 


57 


58 


43.6 


13.6 


84.4 


56.0 


28.4 


01.8 


76.1 


51.3 


27.6 


05.0 


58 


59 


44.8 


14.8 


85.6 


57.2 


29.6 


03.0 


77.3 


52.6 


28.9 


06.3 


59 


60 


1946.0 


2016.0 


JOSO.M 


2158.4 


2230.9 


2304.2 


2378.5 


2453.8 


2530.2 


2607.6 


60 


> i / 

L.J 


30 


31 


32 


33 


34 


35 


36 


37 


38 


39 


' 



Table 5. Meridional Parts 



245 



' 


40 


41 


42 


43 


44 


45 


46 


47 


48 


49 


' 







2607.6 


2686.2 


2766.0 


2847.1 


2929.5 


3013.4 


3098.7 


3185.6 


3274.1 


3364.4 







1 


08.9 


87.6 


67.4 


48.5 


30.9 


14.8 


3100.1 


87.1 


75.6 


65.9 


1 




2 


10.2 


88.9 


68.7 


49.9 


32.3 


16.2 


01.6 


88.5 


77.1 


67.4 


2 




3 


11.5 


90.2 


70.1 


51.2 


33.7 


17.6 


03.0 


90.0 


78.6 


69.0 


3 




4 


12.8 


91.5 


71.4 


52.6 


35.1 


19.0 


04.4 


91.4 


80.1 


70.5 


4 




5 


2614.1 


2692.8 


2772.8 


2853.9 


2936.5 


3020.4 


3105.9 


3192.9 


3281.6 


3372.0 


5 




6 


15.4 


94.2 


74.1 


55.3 


37.9 


21.8 


07.3 


94.4 


83.1 


73.5 


6 




7 


16.8 


95.5 


75.4 


56.7 


39.3 


23.3 


08.8 


95.8 


84.6 


75.1 


7 




8 


18.1 


96.8 


76.8 


58.0 


40.6 


24.7 


10.2 


97.3 


86.1 


76.6 


8 




9 


19.4 


98.1 


78.1 


59.4 


42.0 


26.1 


11.6 


98.8 


87.6 


78.1 


9 




10 


2620.7 


2699.5 


2779.5 


2860.8 


2943.4 


3027.5 


3113.1 


3200.2 


3289.0 


3379.6 


10 




11 


22.0 


2700.8 


80.8 


62.1 


44.8 


28.9 


14.5 


01.7 


90.5 


81.2 


11 




12 


23.3 


02.1 


82.2 


63.5 


46.2 


30.3 


16.0 


03.2 


92.0 


82.7 


12 




13 


24.6 


03.4 


83.5 


64.9 


47.6 


31.7 


17.4 


04.6 


93.5 


84.2 


13 




14 


25.9 


04.8 


84.8 


66.2 


49.0 


33.2 


18.8 


06.1 


95.0 


85.7 


14 




15 


2627.2 


2706.1 


2786.2 


2867.6 


2950.4 


3034.6 


3120.3 


3207.6 


3296.5 


3387.3 


15 




16 


28.5 


07.4 


87.5 


69.0 


51.8 


36.0 


21.7 


09.0 


98.0 


88.8 


16 




17 


29.8 


08.7 


88.9 


70.3 


53.2 


37.4 


23.2 


10.5 


99.5 


90.3 


17 




18 


31.1 


10.1 


90.2 


71.7 


54.5 


38.8 


24.6 


12.0 


3301.0 


91.8 


18 




19 


32.4 


11.4 


91.6 


73.1 


55.9 


40.2 


26.0 


13.4 


02.5 


93.4 


19 




20 


2633.7 


2712.7 


2792.9 


2874.4 


2957.3 


3041.7 


3127.5 


3214.9 


3304.0 


3394.9 


20 




21 


35.0 


14.0 


94.3 


75.8 


58.7 


43.1 


28.9 


16.4 


05.5 


96.4 


21 




22 


36.3 


15.4 


95.6 


77.2 


60.1 


44.5 


30.4 


17.9 


07.0 


98.0 


22 




23 


37.6 


16.7 


97.0 


78.6 


61.5 


45.9 


31.8 


19.3 


08.5 


99.5 


23 




24 


38.9 


18.0 


98.3 


79.9 


62.9 


47.3 


33.3 


20.8 


10.0 


3401.0 


24 




25 


2640.2 


2719.3 


2799.7 


2881.3 


2964.3 


3048.7 


3134.7 


3222.3 


3311.5 


3402.6 


25 




26 


41.6 


20.7 


2801.0 


82.7 


65.7 


50.2 


36.2 


23.7 


13.0 


04.1 


26 




27 


42.9 


22.0 


02.4 


84.0 


67.1 


51.6 


37.6 


25.2 


14.5 


05.6 


27 




28 


44.2 


23.3 


03.7 


85.4 


68.5 


53.0 


39.0 


26.7 


16.0 


07.2 


28 




29 


45.5 


24.7 


05.1 


86.8 


69.9 


54.4 


40.5 


28.2 


17.5 


08.7 


29 




30 


2646.8 


2726.0 


2806.4 


2888.2 


2971.3 


3055.9 


3141.9 


3229.6 


3319.0 


3410.2 


30 




31 


48.1 


27.3 


07.8 


89.5 


72.7 


57.3 


43.4 


31.1 


20.5 


11.8 


31 




32 


49.4 


28.6 


09.1 


90.9 


74.1 


58.7 


44.8 


32.6 


22.1 


13.3 


32 




33 


50.7 


30.0 


10.5 


92.3 


75.5 


60.1 


46.3 


34.1 


23.6 


14.8 


33 




34 


52.0 


31.3 


11.8 


93.7 


76.9 


61.5 


47.7 


35.6 


25.1 


16.4 


34 




35 


2653.3 


2732.6 


2813.2 


2895.0 


2978.3 


3063.0 


3149.2 


3237.0 


3326.6 


3417.9 


35 




36 


54.7 


34.0 


14.5 


96.4 


79.7 


64.4 


50.6 


38.5 


28.1 


19.5 


36 




37 


56.0 


35.3 


15.9 


97.8 


81.1 


65.8 


52.1 


40.0 


29.6 


21.0 


37 




38 


57.3 


36.6 


17.2 


99.2 


82.5 


67.2 


53.5 


41.5 


31.1 


22.5 


38 




39 


58.6 


38.0 


18.6 


2900.5 


83.9 


68.7 


55.0 


42.9 


32.6 


24.1 


39 




40 


2659.9 


2739.3 


2820.0 


2901.9 


2985.3 


3070.1 


3156.4 


3244.4 


3334.1 


3425.6 


40 




41 


61.2 


40.6 


21.3 


03.3 


86.7 


71.5 


57.9 


45.9 


35.6 


27.2 


41 




42 


62.5 


42.0 


22.7 


04.7 


88.1 


72.9 


59.4 


47.4 


37.1 


28.7 


42 




43 


63.9 


43.3 


24.0 


06.1 


89.5 


74.4 


60.8 


48.9 


38.6 


30.2 


43 




44 


65.2 


44.6 


25.'4 


07.4 


90.9 


75.8 


62.3 


50.3 


40.2 


31.8 


44 




45 


2666.5 


2746.0 


2826.7 


2908.8 


2992.3 


3077.2 


3163.7 


3251.8 


3341.7 


3433.3 


45 




46 


67.8 


47.3 


28.1 


10.2 


93.7 


78.7 


65.2 


53.3 


43.2 


34.9 


46 




47 


69.1 


48.6 


29.4 


11.6 


95.1 


80.1 


66.6 


54.8 


44.7 


36.4 


47 




48 


70.4 


50.0 


30.8 


13.0 


96.5 


81.5 


68.1 


56.3 


46.2 


38.0 


48 




49 


71.7 


51.3 


32.2 


14.3 


97.9 


82.9 


69.5 


57.8 


47.7 


39.5 


49 




50 


2673.1 


2752.7 


2833.5 


2915.7 


2999.3 


3084.4 


3171.0 


3259.3 


3349.2 


3441.0 


50 




51 


74.4 


54.0 


34.9 


17.1 


3000.7 


85.8 


72.5 


60.7 


50.8 


42.6 


51 




52 


75.7 


55.3 


36.2 


18.5 


02.1 


87.2 


73.9 


62.2 


52.3 


44.1 


52 




53 


77.0 


56.7 


37.6 


19.9 


03.5 


88.7 


75.4 


63.7 


53.8 


45.7 


53 




54 


78.3 


58.0 


39.0 


21.2 


04.9 


90.1 


76.8 


65.2 


55.3 


47.2 


54 




55 


2679.6 


2759.3 


2840.3 


2922.6 


3006.3 


3091.5 


3178.3 


3266.7 


3356.8 


3448.8 


55 




56 


81.0 


60.7 


41.7 


24.0 


07.7 


93.0 


79.7 


68.2 


58.3 


5Q.3 


56 




57 


82.3 


62.0 


43.0 


25.4 


09.2 


94.4 


81.2 


69.7 


59.9 


51.9 


57 




58 


83.6 


63.4 


44.4 


26.8 


10.6 


95.8 


82.7 


71.1 


61.4 


53.4 


58 




59 


84.9 


64.7 


45.8 


28.2 


12.0 


97.3 


84.1 


72.6 


62.9 


55.0 


59 




60 


2686.2 


2766.0 


2847.1 


2929.5 


3013.4 


3098.7 


3185.6 


3274.1 


3364.4 


3456.5 


60 




' 


40 


41 


42 


43 


44 


45 


46 


47 


48 


49 


,.<* 


m 



246 



Table 5. Meridional Parts 



' 


50 


51 


52 


53 


54 


55 


56 


57 


58 


59 


/ 





3456.5 


3550.6 


3646.7 


3745.1 


3845.7 


3948.8 


4054.5 


4163.0 


4274.4 


4389.1 





1 


58.1 


52.2 


48.4 


46.7 


47.4 


50.5 


56.3 


64.8 


76.3 


91.0 


1 


2 


59.6 


53.8 


50.0 


48.4 


49.1 


52.3 


58.1 


66.6 


78.2 


92.9 


2 


3 


61.2 


55.4 


51.6 


50.0 


50.8 


54.0 


59.8 


68.5 


80.1 


94.9 


3 


4 


62.7 


56.9 


53.2 


51.7 


52.5 


55.7 


61.6 


70.3 


82.0 


96.8 


4 


5 


3464.3 


3558.5 


3654.8 


3753.4 


3854.2 


3957.5 


4063.4 


4172.1 


4283.9 


4398.8 


5 


6 


65.9 


60.1 


56.5 


55.0 


55.9 


59.2 


65.2 


74.0 


85.7 


4400.7 


6 


7 


67.4 


61.7 


58.1 


56.7 


57.6 


61.0 


67.0 


75.8 


87.6 


02.6 


7 


8 


69.0 


63.3 


59.7 


58.3 


59.3 


62.7 


68.8 


77.7 


89.5 


04.6 


8 


9 


70.5 


64.9 


61.3 


60.0 


61.0 


64.5 


70.6 


79.5 


91.4 


06.5 


9 


10 


3472.1 


3566.5 


3663.0 


3761.7 


3862.7 


3966.2 


4072.4 


4181.3 


4293.3 


4408.5 


10 


11 


73.6 


68.1 


64.6 


63.3 


64.4 


68.0 


74.2 


83.2 


95.2 


10.4 


11 


12 


75.2 


69.7 


66.2 


65.0 


66.1 


69.7 


76.0 


85.0 


97.1 


12.4 


12 


13 


76.7 


71.3 


67.9 


66.7 


67.8 


71.5 


77.7 


86.9 


99.0 


14.3 


13 


14 


78.3 


72.8 


69.5 


68.3 


69.5 


73.2 


79.5 


88.7 


4300.9 


16.3 


14 


15 


3479.9 


3574.4 


3671.1 


3770.0 


3871.2 


3975.0 


4081.3 


4190.6 


4302.8 


4418.2 


15 


16 


81.4 


76.0 


72.7 


71.7 


72.9 


76.7 


83.1 


92.4 


04.7 


20.2 


16 


17 


83.0 


77.6 


74.4 


73.3 


74.6 


78.5 


84.9 


94.2 


06.6 


22.1 


17 


18 


84.5 


79.2 


76.0 


75.0 


76.3 


80.2 


86.7 


96.1 


08.5 


24.1 


18 


19 


86.1 


80.8 


77.6 


76.7 


78.1 


82.0 


88.5 


97.9 


10.4 


26.1 


19 


20 


3487.7 


3582.4 


3679.3 


3778.3 


3879.8 


3983.7 


4090.3 


4199.8 


4312.3 


4428.0 


20 


21 


89.2 


84.0 


80.9 


80.0 


81.5 


85.5 


92.1 


4201.6 


14.2 


30.0 


21 


22 


90.8 


85.6 


82.5 


81.7 


83.2 


87.2 


93.9 


03.5 


16.1 


31.9 


22 


23 


92.4 


87.2 


84.2 


83.3 


84.9 


89.0 


95.7 


05.3 


18.0 


33.9 


23 


24 


93.9 


88.8 


85.8 


85.0 


86.6 


90.7 


97.5 


07.2 


19.9 


35.8 


24 


25 


3495.5 


3590.4 


3687.4 


3786.7 


3888.3 


3992.5 


4099.3 


4209.0 


4321.8 


4437.8 


25 


26 


97.1 


92.0 


89.1 


88.4 


90.0 


94.3 


4101.1 


10.9 


23.7 


39.8 


26 


27 


98.6 


93.6 


90.7 


90.0 


91.8 


96.0 


02.9 


12.8 


25.6 


41.7 


27 


28 


3500.2 


95.2 


92.3 


91.7 


93.5 


97.8 


04.8 


14.6 


27.5 


43.7 


28 


29 


01.8 


96.8 


94.0 


93.4 


95.2 


99.5 


06.6 


16.5 


29.4 


45.7 


29 


30 


3503.3 


3598.4 


3695.6 


3795.1 


3896.9 


4001.3 


4108.4 


4218.3 


4331.3 


4447.6 


30 


31 


04.9 


3600.0 


97.3 


96.8 


98.6 


03.1 


10.2 


20.2 


33.2 


49.6 


31 


32 


06.5 


01.6 


98.9 


98.4 


3900.4 


04.8 


12.0 


22.0 


35.2 


51.6 


32 


33 


08.0 


03.2 


3700.5 


3800.1 


02.1 


06.6 


13.8 


23.9 


37.1 


53.5 


33 


34 


09.6 


04.8 


02.2 


01.8 


03.8 


08.3 


15.6 


25.8 


39.0 


55.5 


34 


35 


3511.2 


3606.4 


3703.8 


3803.5 


3905.5 


4010.1 


4117.4 


4227.6 


4340.9 


4457.5 


35 


36 


12.7 


08.0 


05.5 


05.1 


07.2 


11.9 


19.2 


29.5 


42.8 


59.4 


36 


37 


14.3 


09.6 


07.1 


06.8 


09.0 


13.6 


21.0 


31.3 


44.7 


61.4 


37 


38 


15.9 


11.2 


08.7 


08.5 


10.7 


15.4 


22.9 


33.2 


46.6 


63.4 


38 


39 


17.5 


12.8 


10.4 


10.2 


12.4 


17.2 


24.7 


35.1 


48.6 


65.4 


39 


40 


3519.0 


3614.5 


3712.0 


3811.9 


3914.1 


4018.9 


4126.5 


4236.9 


4350.5 


4467.3 


40 


41 


20.6 


16.1 


13.7 


13.6 


15.9 


20.7 


28.3 


38.8 


52.4 


69.3 


41 


42 


22.2 


17.7 


15.3 


15.2 


17.6 


22.5 


30.1 


40.7 


54.3 


71.3 


42 


43 


23.7 


19.3 


17.0 


17.0 


19.3 


24.3 


31.9 


42.5 


56.2 


73.3 


43 


44 


25.3 


20.9 


18.6 


18.6 


21.0 


26.0 


33.8 


44.4 


58.2 


75.3 


44 


45 


3526.9 


3622.5 


3720.3 


3820.3 


3922.8 


4027.8 


4135.6 


4246.3 


4360.1 


4477.2 


45 


46 


28.5 


24.1 


21.9 


22.0 


24.5 


29.6 


37.4 


48.1 


62.0 


79.2 


46 


47 


30.1 


25.7 


23.6 


23.7 


26.2 


31.4 


39.2 


50.0 


63.9 


81.2 


47 


48 


31.6 


27.3 


25.2 


25.4 


28.0 


33.1 


41.0 


51.9 


65.9 


83.2 


48 


49 


33.2 


29.0 


26.9 


27.1 


29.7 


34.9 


42.9 


53.8 


67.8 


85.2 


49 


50 


3534.8 


3630.6 


3728.5 


3828.7 


3931.4 


4036.7 


4144.7 


4255.6 


4369.7 


4487.2 


50 


51 


36.4 


32.2 


30.2 


30.4 


33.2 


38.5 


46.5 


57.5 


71.7 


89.1 


51 


52 


37.9 


33.8 


31.8 


32.1 


34.9 


40.2 


48.3 


59.4 


73.6 


91.1 


52 


53 


39.5 


35.4 


33.5 


33.8 


36.6 


42.0 


50.2 


61.3 


75.5 


93.1 


53 


54 


41.1 


37.0 


35.1 


35.5 


38.4 


43.8 


52.0 


63.1 


77.4 


95.1 


54 


55 


3542.7 


3638.6 


3736.8 


3837.2 


3940.1 


4045.6 


4153.8 


4265.0 


4379.4 


4497.1 


55 


56 


44.3 


40.3 


38.4 


38.9 


41.8 


47.4 


55.7 


66.9 


81.3 


99.1 


56 


57 


45.9 


41.9 


40.1 


40.6 


43.6 


49.1 


57.5 


68.8 


83.2 


4501.1 


57 


58 


47.4 


43.5 


41.7 


42.3 


45.3 


50.9 


59.3 


70.7 


85.2 


03.1 


58 


59 


49.0 


45.1 


43.4 


45.0 


47.0 


52.7 


61.1 


72.5 


87.1 


05.1 


59 


60 


3550.6 


3646.7 


3745.1 


3845.7 


3948.8 


4054.5 


4163.0 


4274.4 


4389.1 


4507.1 


60 


/ 


50 


51 


52 


53 


54 


55 


56 


57 


58 


59 





Table 6 



Table 7 247 



Combined Correction for Observed 
Sextant Altitudes 



Correction for Dip of 

Sea Horizon 

(Sun or Star) 



OBSEHVED 
ALTITUDE 


CORRECTION 


For Sun (to 
be added to 

observed alti- 
tude) 


For Star (to 
be subtracted 

from observed 
altitude) 


5 


6' 14" 


9' 55" 


6 


7 41 


8 28 


7 


8 45 


7 24 


8 


9 35 


6 34 


9 


10 16 


5 53 


10 


10 50 


5 19 


11 


11 17 


4 51 


12 


11 41 


4 27 


13 


12 2 


4 7 


14 


12 19 


3 49 


15 


12 34 


3 34 


20 


13 29 


2 39 


25 


14 3 


2 5 


30 


14 26 


1 41 


35 


14 44 


1 23 


40 


14 57 


1 10 


45 


15 8 


58 


50 


15 17 


49 


55 


15 25 


40 


60 


15 31 


34 


65 


15 37 


27 . 


70 


15 42 


21 


75 


15 47 


16 


80 


15 52 


10 


85 


15 55 


5 



HEIGHT OP 
OBSERVER'S 
EYE ABOVE 
SEA LEVEL 
(feet) 


DIP CORREC- 
TION (to be 
subtracted 

from 
observed 
altitude) 


4 


1' 58" 


6 


2 24 


8 


2 46 


10 


3 06 


12 


3 24 


14 


3 40 


16 


3 55 


18 


4 9 


20 


4 23 


22 


4 36 


24 


4 48 


26 


5 


28 


5 11 


30 


5 22 


35 


5 48 


40 


6 12 


45 


6 36 


50 


6 56 


55 


7 16 


60 


7 35 


70 


8 12 


85 


9 2 


100 


9 48 



Small supplementary correction, for Sun 
only. 

Jan. to March \ jj int . 
and Oct. to Dec. ; add 10 " 
April to Sept., subtract 10". 



The dip correction is not 
required when the artificial 
horizon is used. 



248 



Table 8 



To Change Hours and Minutes into Decimals of a Day 



HOURS EXPRESSED 

AS DECIMAL PARTS 

OF A DAY 



HOURS 


DECIMAL 


1 


.0416 


2 


.0833 


3 


.1250 


4 


.1666 


5 


.2083 


6 


.2500 


7 


.2916 


8 


.3333 


9 


.3750 


10 


.4166 


11 


.4583 


12 


.5000 


13 


.5416 


14 


.5833 


15 


.6249 


16 


.6666 


17 


.7083 


18 


.7500 


19 


.7916 


20 


.8333 


21 


.8749 


22 


.9166 


23 


.9583 


24 


1.0000 



MINUTES EXPRESSED AS DECIMAL PARTS 
OF A DAY 



MINUTES 


DECIMAL 


MINUTES 


DECIMAL 


1 


.0006 


31 


.0215 


2 


.0013 


32 


.0222 


3 


.0020 


33 


.0229 


4 


.0027 


34 


.0236 


5 


.0034 


35 


.0243 


6 


.0041 


36 


.0250 


7 


.0048 


37 


.0256 


8 


.0055 


38 


.0263 


9 


.0062 


39 


.0270 


10 


.0069 


40 


.0277 


11 


.0076 


41 


.0284 


12 


.0083 


42 


.0291 


13 


.0090 


43 


.0298 


14 


.0097 


44 


.0305 


15 


.0104 


45 


.0312 


16 


.0111 


46 


.0319 


17 


.0118 


47 


.0326 


18 


.0125 


48 


.0333 


19 


.0131 


49 


.0340 


20 


.0138 


50 


.0347 


21 


.0145 


51 


.0354 


22 


.0152 


52 


.0361 


23 


.0159 


53 


.0368 


24 


.0166 


54 


.0375 


25 


.0173 


55 


.0381 


26 


.0180 


56 


.0388 


27 


.0187 


57 


.0395 


28 


.0194 


58 


.0402 


29 


.0201 


59 


.0409 


30 


.0208 


bO 


.0416 



Table 9 



249 



To Interchange Degrees and Minutes of Longitude and Hours, Minutes, 
and Seconds of Time. Part 1 





0* 


1* 


2* 


3* 


4A 


6* 


6* 


7* 


8* 


9* 


10* 


11* 


o m 





15 


30 


45 


60 


75 


90 


105 


120 


135 


150 


165 


4 


1 


16 


31 


46 


61 


76 


91 


106 


121 


136 


151 


166 


8 


2 


17 


32 


47 


62 


77 


92 


107 


122 


137 


152 


167 


12 


3 


18 


33 


48 


63 


78 


93 


108 


123 


138 


153 


168 


16 


4 


19 


34 


49 


64 


79 


94 


109 


124 


139 


154 


169 


20 


5 


20 


35 


50 


65 


80 


95 


110 


125 


140 


155 


170 


24 


6 


21 


36 


51 


66 


81 


96 


111 


126 


141 


156 


171 


28 


7 


22 


37 


52 


67 


82 


97 


112 


127 


142 


157 


172 


32 


8 


23 


38 


53 


68 


83 


98 


113 


128 


143 


158 


173 


36 


9 


24 


39 


54 


69 


84 


99 


114 


129 


144 


159 


174 


40 


10 


25 


40 


55 


70 


85 


100 


115 


130 


145 


160 


175 


44 


11 


26 


41 


56 


71 


86 


101 


116 


131 


146 


161 


176 


48 


12 


27 


42 


57 


72 


87 


102 


117 


132 


147 


162 


177 


52 


13 


28 


43 


58 


73 


88 


103 


118 


133 


148 


163 


178 


56 


14 


29 


44 


59 


74 


89 


104 


119 


134 


149 


164 


179 






12* 


13* 


14* 


f5* 


16* 


17* 


18* 


19* 


20* 


21* 


22* 


23* 


0" ! 


180 


195 


210 


225 


240 


255 


270 


285 


300 


315 


330 


345 


4 


181 


196 


211 


226 


241 


256 


271 


286 


301 


316 


331 


346 


8 


182 


197 


212 


227 


242 


257 


272 


287 


302 


317 


332 


347 


12 


183 


198 


213 


228 


243 


258 


273 


288 


303 


318 


333 


348 


16 


184 


199 


214 


229 


244 


259 


274 


289 


304 


319 


334 


349 


20 


185 


200 


215 


230 


245 


260 


275 


290 


305 


320 


335 


350 


24 


186 


201 


216 


231 


246 


261 


276 


291 


306 


321 


336 


351 


28 


187 


202 


217 


232 


247 


262 


277 


292 


307 


322 


337 


352 


32 


188 


203 


218 


233 


248 


263 


278 


293 


308 


323 


338 


353 


36 


189 


204 


219 


234 


249 


264 


279 


294 


309 


324 


339 


354 


40 


190 


205 


220 


235 


250 


265 


280 


295 


310 


325 


340 


355 


44 


191 


206 


221 


236 


251 


266 


281 


296 


311 


326 


341 


356 


48 


192 


207 


222 


237 


252 


267 


282 


297 


312 


327 


342 


357 


52 


193 


208 


223 


238 


253 


268 


283 


298 


313 


328 


343 


358 


56 


194 


209 


224 


239 


254 


269 


284 


299 


314 


329 


344 


359 



Part 2 



EXPLANATION OP TABLE 9 

1. To change degrees of longitude into hours and 
minutes of time : Find the number of degrees in Part 1. 
The required hours will then be found at the head of the 
column containing the degrees, and the required min- 
utes at the left-hand end of the line containing the 
degrees. 

Examples: 113 = 7* 32 m ; 294 = 19* 36 m . 

2. To change minutes of longitude into minutes and 
seconds of time : Find the minutes of longitude in Part 2. 
The required minutes and seconds of time will again 
be found at the head of the column and the left-hand end 
of the line. 

Examples : 43' = 2 m 52 s ; 28' = l m 52". 

3. 1 and 2 can be combined by addition. 

Examples : 113 43' = 7* 34 m 52 s . 
294 28' = 19* 37 m 52. 

4. To change hours and minutes of time into degrees 
and minutes of longitude : Find the number of hours at 
the head of one of the columns of Part 1 ; then run down 
the column until you reach a line having at its left-hand 
end a number of minutes equal to (or just smaller than) 
the given number of minutes of time. Where that line 

and column meet you will find the required degrees of longitude. 

Examples: 7' 32 m = 113; 19* 36 m = 294. 

5. To change minutes and seconds of time into minutes of longitude : Find the number of 
minutes of time at the head of one of the columns of Part 2 ; then run down the column until 
you reach a line having at its left-hand end a number of seconds equal (or nearly equal) to 
the given number of seconds of time. Where that line and column meet you will find the 
minutes of longitude. 

Examples : 2 m 52* = 43' ; l m 52 s = 28'. 

6. 4 and 5 can be combined by addition : 

Examples : 7 34 m 52' = 1 13 43' ; 19* 37 m 52* = 294 28'. 





Qm 


1"' 


2 m 


8 


s 


0' 


15' 


30' 


45' 


4 


1 


16 


31 


46 


8 


2 


17 


32 


47 


12 


3 


18 


33 


48 


16 


4 


19 


34 


49 


20 


5 


20 


35 


50 


24 


6 


21 


36 


51 


28 


7 


22 


37 


52 


32 


8 


23 


38 


53 


36 


9 


24 


39 


54 


40 


10 


25 


40 


55 


44 


11 


26 


41 


56 


48 


12 


27 


42 


57 


52 


13 


28 


43 


58 


56 


14 


29 


44 


59 



250 



Table 10. Haversine Table 



s ' 


O h O m 


Oh 4 1 


Oh s 2 


Qh 1S m 30 




1I.IV. 


No. 


Hav. 


No. 


Hav. 


No. 


Bar. 


No. 







0.00000 


5.88168 


0.00008 


6.48371 


0.00030 


6.83584 


0.00069 


4 1 


2.32539 


.00000 


.89604 


.00008 


.49092 


.00031 


.84065 


.00069 


8 2 


.92745 


.00000 


.91016 


.00008 


.49807 


.00031 


.84543 


.00070 


12 3 


3.27963 


.00000 


.92406 


.00008 


.50516 


.00032 


.85019 


.00071 


16 4 


.52951 


.00000 


.93774 


.00009 


.51219 


.00033 


.85492 


.00072 


20 5 


3.72333 


0.00000 


5.95121 


0.00009 


6.51916 


0.00033 


6.85963 


0.00072 


24 6 


.88169 


.00000 


.96447 


.00009 


.52608 


.00034 


.86431 


.00073 


28 7 


4.01559 


.00000 


.97753 


.00010 


.53295 


.00034 


.86897 


.00074 


32 8 


.13157 


.00000 


.99040 


.00010 


.53976 


.00035 


.87360 


.00075 


36 9 


.23388 


.00000 


6.00308 


.00010 


.54652 


.00035 


.87821 


.00076 


40 10 


4.32539 


0.00000 


6.01557 


0.00010 


6.55323 


0.00036 


6.88279 


0.00076 


44 11 


.40818 


.00000 


.02789 


.00011 


.55988 


.00036 


.88735 


.00077 


48 12 


.48375 


.00000 


.04004 


.00011 


.56649 


.00037 


.89188 


.00078 


52 13 


.55328 


.00000 


.05202 


.00011 


.57304 


.00037 


.89639 


.00079 


56 14 


.61765 


.00000 


.06384 


.00012 


.57955 


.00038 


.90088 


.00080 


s ' 


Qh jm QO 


Oh 6 >n jo 


Qhgm 2 


O h 13 m 3 


15 


4.67757 


0.00000 


6.07550 


0.00012 


6.58600 


0.00039 


6.90535 


0.00080 


4 16 


.73363 


.00001 


.08700 


.00012 


.59241 


.00039 


.90979 


.00081 


S 17 


.78629 


.00001 


.09836 


.00013 


.59878 


.00040 


.91421 


.00082 


12 18 


.83594 


.00001 


.10956 


.00013 


.60509 


.00040 


.91860 


.00083 


76 19 


.88290 


.00001 


.12063 


.00013 


.61136 


.00041 


.92298 


.00084 


20 20 


4.92745 


0.00001 


6.13155 


0.00014 


6.61759 


0.00041 


6.92733 


0.00085 


24 21 


.96983 


.00001 


.14234 


.00014 


.62377 


.00042 


.93166 


.00085 


2S 22 


5.01024 


.00001 


.15300 


.00014 


.62991 


.00043 


.93597 


.00086 


32 23 


.04885 


.00001 


.16353 


.00015 


.63600 


.00043 


.94026 


.00087 


36 24 


.08581 


.00001 


.17393 


.00015 


.64205 


.00044 


.94453 


.00088 


40 25 


5.12127 


0.00001 


6.18421 


0.00015 


6.64806 


0.00044 


6.94877 


0.00089 


44 26 


.15534 


.00001 


.19437 


.00016 


.65403 


.00045 


.95300 


.00090 


45 27 


.18812 


.00002 


.20441 


.00016 


.65996 


.00046 


.95720 


.00091 


52 28 


.21971 


.00002 


.21433 


.00016 


.66585 


.00046 


.96139 


.00091 


56 29 


.25019 


.00002 


.22415 


.00017 


.67170 


.00047 


.96555 


.00092 


s ' 


Qh 2 m QO 


Qhffm jo 


Oh io m 2 


Oh 14 3 


0- 30 


5.27963 


0.00002 


6.23385 


0.00017 


6.67751 


0.00048 


6.96970 


0.00093 


4 31 


.30811 


.00002 


.24345 


.00018 


.68328 


.00048 


.97382 


.00094 


8 32 


.33569 


.00002 


.25294 


.00018 


.68901 


.00049 


.97793 


.00095 


72 33 


.36242 


.00002 


.26233 


.00018 


.69470 


.00050 


.98201 


.00096 


76 34 


.38835 


.00002 


.27162 


.00019 


.70036 


.00050 


.98608 


.00097 


20 35 


5.41352 


0.00003 


6.28081 


0.00019 


6.70598 


0.00051 


6.99013 


0.00098 


24 36 


.43799 


.00003 


.28991 


.00019 


.71157 


.00051 


.99416 


.00099 


28 37 


.46179 


.00003 


.29891 


.00020 


.71712 


.00052 


.99817 


.00100 


32 38 


.48496 


.00003 


.30781 


.00020 


.72263 


.00053 


7.00216 


.00101 


3(5 39 


.50752 


.00003 


.31663 


.00021 


.72811 


.00053 


.00613 


.00101 


40 40 


5.52951 


0.00003 


6.32536 


0.00021 


6.73355 


0.00054 


7.01009 


0.00102 


44 41 


.55095 


.00004 


.33400 


.00022 


.73896 


.00055 


.01403 


.00103 


48 42 


.57189 


.00004 


.34256 


.00022 


.74434 


.00056 


.01795 


.00104 


52 43 


.59232 


.00004 


.35103 


.00022 


.74969 


.00056 


.02185 


.00105 


56 44 


.61229 


.00004 


.35943 


.00023 


.75500 


.00057 


02573 


.00106 


s ' 


Qh 3 m 00 


Qh 7 m JO 


0*11 2 


Qh 15 m 3 


45 


5.63181 


0.00004 


6.36774 


0.00023 


6.76028 


0.00058 


7.02960 


0.00107 


4 46 


.65090 


.00004 


.37597 


.00024 


.76552 


.00058 


.03345 


.00108 


5 47 


.66958 


.00005 


.38412 


.00024 


.77074 


.00059 


.03729 


.00109 


/2 48 


.68787 


.00005 


.39220 


.00025 


.77592 


.00060 


.04110 


.00110 


16 49 


.70578 


.00005 


.40021 


.00025 


.78108 


.00060 


.04490 


.00111 


20 50 


5.72332 


0.00005 


6.40814 


0.00026 


6.78620 


0.00061 


7.04869 


0.00112 


24 51 


.74052 


.00006 


.41600 


.00026 


.79129 


.00062 


.05245 


.00113 


28 52 


.75739 


.00006 


.42379 


.00027 


.79630 


.00063 


.05620 


.00114 


32 53 


.77394 


.00006 


.43151 


.00027 


.80139 


.00063 


.05994 


.00115 


36 54 


.79017 


.00006 


.43916 


.00027 


.80640 


.00064 


.06366 


.00116 


40 55 


5.80611 


0.00006 


6.44675 


0.00028 


6.81137 


0.00065 


7.06736 


0.00117 


44 56 


.82176 


.00007 


.45427 


.00028 


.81632 


.00066 


.07105 


.00118 


45 57 


.83713 


.00007 


.46172 


.00029 


.82124 


.00066 


.07472 


.00119 


52 58 


.85224 


.00007 


.46911 


.00029 


.82614 


.00067 


.07837 


.00120 


56 59 


.86709 


.00007 


.47644 


.00030 


.83100 


.00068 


.08201 


.00121 


<?0 60 


5.88168 


0.00008 


6.48371 


0.00030 


6.83584 


0.00069 


7.08564 


0.00122 



Table 10. Hayersine Table 



251 



S ' 


0" 16 4 


0* 20 m 5 


0A 24 m 6 . 


0* 28 7 




Hav. 


No. 


Hav. 


No. 


Hav. 


No. 


Hav. 


No. 





7.08564 


0.00122 


7.27936 


0.00190 


7.43760 


0.00274 


7.57135 


0.00373 


4 l 


.08925 


.00123 


.28225 


.00192 


.44001 


.00275 


.57341 


.00374 | 


8 2 


.09284 


.00124 


.28513 


.00193 


.44241 


.00277 


.57547 


.00376 


12 3 


.09642 


.00125 


.28800 


.00194 


.44480 


.00278 


.57752 


.00378 


16 4 


.09999 


.00126 


.29086 


.00195 


.44719 


.00280 


.57957 


.00380 


20 5 


7.10354 


0.00127 


7.29371 


0.00197 


7.44957 


0.00282 


7.58162 


0.00382 


24 6 


.10708 


.00128 


.29655 


.00198 


.45194 


.00283 


.58366 


.00383 


28 7 


.11060 


.00129 


.29938 


.00199 


.45431 


.00285 


.58569 


.00385 


32 8 


.11411 


.00130 


.30220 


.00201 


.45667 


.00286 


.58772 


.00387 


36 9 


.11760 


.00131 


.30502 


.00202 


.45903 


.00288 


.58974 


.00389 


40 10 


7.12108 


0.00132 


7.30782 


0.00203 


7.46138 


0.00289 


7.59176 


0.00391 


44 11 


.12455 


.00133 


.31062 


.00204 


.46372 


.00291 


.59378 


.00392 


48 12 


.12800 


.00134 


.31340 


.00206 


.46605 


.00292 


.59579 


.00394 


52 13 


.13144 


.00135 


.31618 


.00207 


.46838 


.00294 


.59779 


.00396 


56 14 


.13486 


.00136 


.31895 


.00208 


.47071 


.00296 


.59979 


.00398 


s ' 


Qh 17 m 4 


Qh 2/m 5 


0* 25 6 


Oh 29 7 


15 


7.13827 


0.00137 


7.32171 


0.00210 


7.47302 


0.00297 


7.60179 


0.00400 


4 16 


.14167 


.00139 


.32446 


.00211 


.47533 


.00299 


.60378 


.00402 


5 17 


.14506 


.00140 


.32720 


.00212 


.47764 


.00300 


.60577 


.00403 


12 18 


.14843 


.00141 


.32994 


.00214 


.47994 


.00302 


.60775 


.00405 


1(5 19 


.15179 


.00142 


.33266 


.00215 


.48223 


.00304 


.60973 


.00407 


20 


7.15513 


0.00143 


7.33538 


0.00216 


7.48452 


0.00305 


7.61170 


0.00409 


24 21 


.15846 


.00144 


.33809 


.00218 


.48680 


.00307 


.61367 


.00411 


5 22 


.16178 


.00145 


.34079 


.00219 


.48907 


.00308 


.61564 


.00413 


32 23 


.16509 


.00146 


.34348 


.00221 


.49134 


.00310 


.61760 


.00415 


36 24 


.16839 


.00147 


.34616 


.00222 


.49360 


.00312 


.61955 


.00416 


40 25 


7.17167 


0.00148 


7.34884 


0.00223 


7.49586 


0.00313 


7.62151 


0.00418 


44 26 


.17494 


.00150 


.35150 


.00225 


.49811 


.00315 


.62345 


.00420 


45 27 


.17820 


.00151 


.35416 


.00226 


.50036 


.00316 


.62540 


.00422 


52 28 


.18144 


.00152 


.35681 


.00227 


.50259 


.00318 


.62733 


.00424 


55 29 


.18468 


.00153 


.35945 


.00229 


.50483 


.00320 


.62927 


.00426 


s ' 


0* 18 4 


0*22 5 


Oh 26 6 


0*30 7 


30 


7.18790 


0.00154 


7.36209 


0.00230 


7.50706 


0.00321 


7.63120 


0.00428 


4 31 


.19111 


.00155 


.36471 


.00232 


.50928 


.00323 


.63312 


.00430 


8 32 


.19430 


.00156 


.36733 


.00233 


.51149 


.00325 


.63504 


.00432 


.72 33 


.19749 


.00158 


.36994 


.00234 


.51370 


.00326 


.63696 


.00433 


/'/ 34 


.20066 


.00159 


.37254 


.00236 


.51591 


.00328 


.63887 


.00435 


20 35 


7.20383 


0.00160 


7.37514 


0.00237 


7.51811 


0.00330 


7.64078 


0.00437 


24 36 


.20698 


.00161 


.37773 


.00239 


.52030 


.00331 


.64269 


.00439 


28 37 


.21012 


.00162 


.38030 


.00240 


.52249 


.00333 


.64458 


.00441 


32 38 


.21325 


.00163 


.38288 


.00241 


.52467 


.00335 


.64648 


.00443 


36 39 


.21636 


.00165 


.38544 


.00243 


.52685 


.00336 


.64837 


.00445 


40 40 


7.21947 


0.00166 


7.38800 


0.00244 


7.52902 


0.00338 


7.65026 


0.00447 


44 41 


.22256 


.00167 


.39054 


.00246 


.53119 


.00340 


.65214 


.00449 


48 42 


.22565 


.00168 


.39309 


.00247 


.53335 


.00341 


.65402 


.00451 


52 43 


.22872 


.00169 


.39562 


.00249 


.53550 


.00343 


.65590 


.00453 


56 44 


.23178 


.00171 


.39815 


.00250 


.53766 


.00345 


.65777 


.00455 


s ' 


0* 1ST 4 


0*23 5 


Oh 27 m 6 


Oh si m 7 


45 


7.23483 


0.00172 


7.40067 


0.00252 


7.53980 


0.00347 


7.65964 


0.00457 


4 46 


.23787 


.00173 


.40318 


.00253 


.54194 


.00348 


.66150 


.00459 


S 47 


.24090 


.00174 


.40568 


.00255 


.54407 


.00350 


.66336 


.00461 


J2 48 


.24392 


.00175 


.40818 


.00256 


.54620 


.00352 


.66521 


.00463 


16 49 


.24693 


.00177 


.41067 


.00257 


.54833 


.00353 


.66706 


.00465 


20 50 


7.24993 


0.00178 


7.41315 


0.00259 


7.55045 


0.00355 


7.66891 


0.00467 


24 51 


.25292 


.00179 


.41563 


.00260 


.55256 


.00357 


.67075 


00469 


2S 52 


.25590 


.00180 


.41810 


.00262 


.55467 


.00359 


.67259 


.00471 


{32 53 


.25886 


.00181 


.42056 


.00263 


.55677 


.00360 


.67443 


.00473 


36 54 


.26182 


.00183 


.42301 


.00265 


.55887 


.00362 


.67626 


.00475 


40 55 


7.26477 


0.00184 


7.42546 


0.00266 


7.56096 


0.00364 


7.67809 


0.00477 


44 56 


.26771 


.00185 


.42790 


.00268 


.56305 


.00366 


.67991 


.00479 


48 57 


.27064 


.00186 


.43034 


.00269 


.56513 


.00367 


.68173 


.00481 


52 58 


.27355 


.00188 


.43277 


.00271 


.56721 


.00369 


.68355 


.00483 


56 59 


.27646 


.00189 


.43519 


.00272 


.56928 


.00371 


.68536 


.00485 


60 60 


7.27936 


0.00190 


7.43760 


0.00274 


7.57135 


0.00373 


7.68717 


0.00487 



252 



Table 10. Haversine Table 



s ' 


O h 32 m 8 


O h 36 9 


Qh 40 m 10 


Oh 44 m 11 




Hav. 


No. 


Hav. 


No. 


Hav. 


No. 


Hav. 


No. 





7.68717 


0.00487 


7.78929 


0.00616 


7.88059 


0.00760 


7.96315 


0.00919 


4 1 


.68897 


.00489 


.79089 


.00618 


.88203 


.00762 


.96446 


.00921 


8 2 


.69077 


.00491 


.79249 


.00620 


.88348 


.00765 


.96577 


.00924 


12 3 


.69257 


.00493 


.79409 


.00622 


.88491 


.00767 


.96707 


.00927 


16 4 


.69437 


.00495 


.79568 


.00625 


.88635 


.00770 


.96838 


.00930 


20 5 


7.69616 


0.00497 


7.79728 


0.00627 


7.88778 


0.00772 


7.96968 


0.00933 


24 6 


.69794 


.00499 


.79886 


.00629 


.88921 


.00775 


.97098 


.00935 


28 7 


.69972 


.00501 


.80045 


.00632 


.89064 


.00777 


.97228 


.00938 


32 8 


.70150 


.00503 


.80203 


.00634 


.89207 


.00780 


.97358 


.00941 


36 9 


.70328 


.00505 


.80361 


.00636 


.89349 


.00783 


.97478 


.00944 


40 10 


7.70505 


0.00507 


7.80519 


0.00639 


7.89491 


0.00785 


7.97617 


0.00947 


44 11 


.70682 


.00509 


.80677 


.00641 


.89633 


.00788 


.97746 


.00949 


48 12 


.70858 


.00511 


.80834 


.00643 


.89775 


.00790 


.97875 


00952 


52 13 


.71034 


.00513 


.80991 


.00646 


.89916 


.00793 


.98003 


.00955 


f>6 14 


.71210 


.00515 


.81147 


.00648 


.90057 


.00795 


.98132 


.00958 


s ' 


Qh 33m 8 


O h 37 m 90 


Qh 41 10 


0^ 45 m 11 


f 15 


7.71385 


0.00517 


7.81303 


0.00650 


7.90198 


0.00798 


7.98260 


0.00961 


4 16 


.71560 


.00520 


.81459 


.00653 


.90339 


.00801 


.98389 


.00964 


/? 17 


.71735 


.00522 


.81615 


.00655 


.90480 


.00803 


.98517 


.00966 


12 18 


.71909 


.00524 


.81771 


.00657 


.90620 


.00806 


.98644 


.00969 


/6 19 


.72083 


.00526 


,81926 


.00660 


.90760 


.00808 


.98772 


.00972 


20 


7.72257 


0.00528 


7.82081 


0.00662 


7.90900 


0.00811 


7.98899 


0.00975 


24 21 


.72430 


.00530 


.82235 


.00664 


.91039 


.00814 


.99027 


.C0978 


2S 22 


.72603 


.00532 


.82390 


.00667 


.91179 


.00816 


.99154 


.00981 


32 23 


.72775 


.00534 


.82544 


.00669 


.91318 


.00819 


.99281 


.00984 


3<? 24 


.72948 


.00536 


.82698 


.00671 


.91457 


.00821 


.99407 


.00986 


40 25 


7.73119 


0.00539 


7.82851 


0.00674 


7.91596 


0.00824 


7.99534 


0.00989 


44 26 


.73291 


.00541 


.83004 


.00676 


.91734 


.00827 


.99660 


.00992 


45 27 


.73462 


.00543 


.83157 


.00679 


.91872 


.00829 


.99786 


.00995 


52 28 


.73633 


.00545 


.83310 


.00681 


.92010 


.00832 


.99912 


.00998 


5<J 29 


.73803 


.00547 


.83463 


.00683 


.92148 


.00835 


8.00038 


.01001 


s ' 


Oh 34 m 8 


Oh $8 9 


0*42 10 


0* 46 11 


30 


7.73974 


0.00549 


7.83615 


0.00686 


7.92286 


0.00837 


8.00163 


0.01004 


4 31 


.74143 


.00551 


.83767 


.00688 


.92423 


.00840 


.00289 


.01007 


8 32 


.74313 


.00554 


.83918 


.00691 


.92560 


.00843 


.00414 


.01010 


J2 33 


.74482 


.00556 


.84070 


.00693 


.92697 


.00845 


.00539 


.01012 


itf 34 


.74651 


.00558 


.84221 


.00695 


.92834 


.00848 


.00664 


.01015 


20 35 


7.74819 


0.00560 


7.84372 


0.00698 


7.92970 


0.00851 


8.00788 


0.01018 


24 36 


.74988 


.00562 


.84522 


.00700 


.93107 


.00853 


.00913 


.01021 


28 37 


.75155 


.00564 


.84672 


.00703 


.93243 


.00856 


.01037 


.01024 


32 38 


.75323 


.00567 


.84822 


.00705 


.93379 


.00859 


.01161 


.01027 


3<S 39- 


.75490 


.00569 


.84972 


.00707 


.93514 


.00861 


.01285 


.01030 


40 40 


7.75657 


0.00571 


7.85122 


0.00710 


7.93650 


0.00864 


8.01409 


0.01033 


44 41 


.75824 


.00573 


.85271 


.00712 


.93785 


.00867 


.01532 


.01036 


48 42 


.75990 


.00575 


.85420 


.00715 


.93920 


.00869 


.01656 


.01039 


52 43 


.76156 


.00578 


.85569 


.00717 


.94055 


.00872 


.01779 


.01042 


50 44 


.76321 


.00580 


.85717 


.00720 


.94189 


.00875 


.01902 


.01045 


s ' 


Oh 35 m 8 


0*35 9 


Oh 43 10 


Oh 47 m 11 


45 


7.76487 


0.00582 


7.85866 


0.00722 


7.94324 


0.00877 


8.02025 


0.01048 


4 46 


.76652 


.00584 


.86014 


.00725 


.94458 


.00880 


.02148 


.01051 


S 47 


.76816 


.00586 


.86161 


.00727 


.94592 


.00883 


.02270 


.01054 


/.' 48 


.76981 


.00589 


.86309 


.00730 


.94726 


.00886 


.02392 


.01057 


16 49 


.77145 


.00591 


.86456 


.00732 


.94859 


.00888 


.02515 


.01060 


20 50 


7.77308 


0.00593 


7.86603 


0.00735 


7.94992 


0.00891 


8.02637 


0.01063 


24 51 


.77472 


.00595 


.86750 


.00737 


.95126 


.00894 


.02758 


.01066 


25 52 


.77635 


.00598 


.86896 


.00740 


.95259 


.00897 


.02880 


.01069 


32 53 


.77798 


.00600 


.87042 


.00742 


.95391 


.00899 


.03001 


.01072 


36 54 


.77960 


.00602 


.87188 


.00745 


.95524 


.00902 


.03123 


.01075 


40 55 


7.78122 


0.00604 


7.87334 


0.00747 


7.95656 


0.00905 


8.03244 


0.01078 


44 56 


.78284 


.00607 


.87480 


.00750 


.95788 


.00908 


.03365 


.01081 


45 57 


.78446 


.00609 


.87625 


.00752 


.95920 


.00910 


.03486 


.01084 


52 58 


.78607 


.00611 


.87770 


.00755 


.96052 


.00913 


.03606 


.01087 


56 59 


.78768 


.00613 


.87915 


.00757 


.96183 


.00916 


.03727 


.01090 


60 60 


7.78929 


0.00616 


7.88059 


0.00760 


7.96315 


0.00919 


8.03847 


001093 



Table 10. Haversine Table 



253 



s ' 


0* 4#"' 12 


0A 52 m 13 


0A 56 14 


lh (jm. 150 




Hav. 


No. 


Hav. 


No. 


Hav. 


No. 


Hav. 


No. 





8.03847 


0.01093 


8.10772 


0.01282 


8.17179 


0.01485 


8.23140 


0.01704 


4 1 


.03967 


.01096 


.10883 


.01285 


.17282 


.01489 


.23235 


.01707 


8 2 


.04087 


.01099 


.10993 


.01288 


.17384 


.01492 


.23331 


.01711 


12 3 


.04207 


.01102 


.11104 


.01291 


.17487 


.01496 


.23427 


.01715 


16 4 


.04326 


.01105 


.11214 


.01295 


.17590 


.01499 


.23523 


.01719 


20 5 


8.04446 


0.01108 


8.11324 


0.01298 


8.17692 


0.01503 


8.23618 


0.01723 


24 6 


.04505 


.01111 


.11435 


.01301 


.17794 


.01506 


.23713 


.01726 


28 1 


.04684 


.01114 


.11544 


.01305 


.17896 


.01510 


.23809 


.01730 


3> 8 


.04803 


.01117 


.11654 


.01308 


.17998 


.01513 


.23904 


.01734 


36 9 


.04922 


.01120 


.11764 


.01311 


.18100 


.01517 


.23999 


.01738 


40 10 


8.05041 


0.01123 


8.11873 


0.01314 


8.18202 


0.01521 


8.24094 


0.01742 


44 11 


.05159 


.01126 


.11983 


.01317 


.18303 


.01524 


.24189 


.01745 


48 12 


.05277 


.01129 


.12092 


.01321 


.18405 


.01528 


.24283 


.01749 


52 13 


.05395 


.01132 


.12201 


.01324 


.18506 


.01531 


.24378 


.01753 


56 14 


.05513 


.01135 


.12310 


.01328 


.18607 


.01535 


.24473 


.01757 


s ' 


0* A9 m 12 


0ft 53" 1 13 


Oh 57 m 14 


!h jm 15 


15 


8.05631 


0.01138 


8.12419 


0.01331 


8.18709 


0.01538 


8.24567 


0.01761 


4 16 


.05749 


.01142 


.12528 


.01334 


.18810 


.01542 


.24661 


.01764 


S 17 


.05866 


.01145 


.12636 


.01338 


.18910 


.01546 


.24755 


.01768 


12 18 


.05984 


.01148 


.12745 


.01341 


.19011 


.01549 


.24850 


.01772 


10 19 


.06101 


.01151 


.12853 


.01344 


.19112 


.01553 


.24944 


.01776 


20 20 


8.06218 


0.01154 


8.12961 


0.01348 


8.19212 


0.01556 


8.25037 


0.01780 


24 21 


.06335 


.01157 


.13069 


.01351 


.19313 


.01560 


.25131 


.01784 


2S 22 


.06451 


.01160 


.13177 


.01354 


.19413 


.01564 


.25225 


.01788 


32 23 


.06568 


.01163 


.13285 


.01358 


.19513 


.01567 


.25319 


.01791 


36 24 


.06684 


.01166 


.13392 


.01361 


.19613 


.01571 


.25412 


.01795 


40 25 


8.06800 


0.01170 


8.13500 


0.01365 


8.19713 


0.01574 


8.25505 


0.01799 


44 26 


.06917 


.01173 


.13607 


.01368 


.19813 


.01578 


.25599 


.01803 


45 27 


.07032 


.01176 


.13714 


.01371 


.19913 


.01582 


.25692 


.01807 


52 28 


.07148 


.01179 


.13822 


.01375 


.20012 


.01585 


.25785 


.01811 


5(5 29 


.07264 


.01182 


.13928 


.01378 


.20112 


.01589 


.25878 


.01815 


s ' 


0* 50" 12 


0* 54"* 13 


0* 68 14 


lh m 15 


30 


8.07379 


0.01185 


8.14035 


0.01382 


8.20211 


0.01593 


8.25971 


0.01818 


4 31 


.07494 


.01188 


.14142 


.01385 


.20310 


.01596 


.26064 


.01822 


8 32 


.07610 


.01192 


.14248 


.01388 


.20410 


.01600 


.26156 


.01826 


J2 33 


.07725 


.01195 


.14355 


.01392 


.20509 


.01604 


.26249 


.01830 


/6 34 


.07839 


.01198 


.14461 


.01395 


.20608 


.01607 


.26341 


.01834 


20 35 


8.07954 


0.01201 


8.14567 


0.01399 


8.20706 


0.01611 


8.26434 


0.01838 


24 36 


.08069 


.01204 


.14673 


.01402 


.20805 


.01615 


.26526 


.01842 


28 37 


.08183 


.01207 


.14779 


.01405 


.20904 


.01618 


.26618 


.01846 


32 38 


.08297 


.01211 


.14885 


.01409 


.21002 


.01622 


.26710 


.01850 


36 39 


.08411 


.01214 


.14991 


.01412 


.21100 


.01626 


.26802 


.01854 


40 40 


8.08525 


0.01217 


8.15096 


0.01416 


8.21199 


0.01629 


8.26894 


0.01858 


4-4 41 


. .08639 


.01220 


.15201 


.01419 


.21297 


.01633 


.26986 


.01861 


48 42 


.08752 


.01223 


.15307 


.01423 


.21395 


.01637 


.27078 


.01865 


52 43 


.08866 


.01226 


.15412 


.01426 


.21493 


.01640 


.27169 


.01869 


56 44 


.08979 


.01230 


.15517 


.01429 


.21590 


.01644 


.27261 


.01873 


8 ' 


0* 51 m 12 


0A 55 m 13 


0* 59 m 14 


lh S 15 


45 


8.09092 


0.01233 


8.15622 


0.01433 


8.21688 


0.01648 


8.27352 


0.01877 


4 46 


.09205 


.01236 


.15726 


.01436 


.21785 


.01651 


.27443 


.01881 


5 47 


.09318 


.01239 


.15831 


.01440 


.21883 


.01655 


.27534 


.01885 


/# 48 


.09431 


.01243 


.15935 


.01443 


.21980 


.01659 


.27626 


.01889 


16 49 


.09543 


.01246 


.16040 


.01447 


.22077 


.01663 


.27717 


.01893 


20 50 


8.09656 


0.01249 


8.16144 


0.01450 


8.22175 


0.01666 


8.27807 


0.01897 


24 51. 


.09768 


.01252 


.16248 


.01454 


.22272 


.01670 


.27898 


.01901 


2S 52 


.09880 


.01255 


.16352 


.01457 


.22368 


.01674 


.27989 


.01905 


32 53 


.09992 


.01259 


.16456 


.01461 


.22465 


.01677 


.28080 


.01909 


36 54 


.10104 


.01262 


.16559 


.01464 


.22562 


.01681 


.28170 


.01913 


40 55 


8.10216 


0.01265 


8.16663 


0.01468 


8.22658 


0.01685 


8.28260 


0.01917 


44 56 


.10327 


.01268 


.16766 


.01471 


.22755 


.01689 


.28351 


.01921 


48 57 


.10439 


.01272 


.16870 


.01475 


.22851 


.01692 


.28441 


.01925 


52 58 


.10550 


.01275 


.16973 


.01478 


.22947 


.01696 


.28531 


.01929 


56 59 


.10661 


.01278 


.17076 


.01482 


.23044 


.01700 


.28621 


.01933 


60 60 


8.10772 


0.01282 


8.17179 


0.01485 


8.23140 


0.01704 


8.28711 


0.01937 



254 



Table 10. Haversine Table 



s ' 


1* 4 m 16 


Ik 8 m 17 


Ik 12 18 


l h 16 19 




Hav. 


No. 


Hav. 


No. 


Uav. 


No. 


Hav. 


No. 





8.28711 


0.01937 


8.33940 


0.02185 


8.38867 


0.02447 


8.43522 


0.02724 


4 1 


.28801 


.01941 


.34025 


.02189 


.38946 


.02452 


.43597 


.02729 


8 2 


.28891 


.01945 


.34109 


.02193 


.39026 


.02456 


.43673 


.02734 


13 3 


.28980 


.01949 


.34194 


.02198 


.39105 


.02461 


.43748 


.02738 


16 4 


.29070 


.01953 


.34278 


.02202 


.39185 


.02465 


.43823 


.02743 


20 5 


8.29159 


0.01957 


8.34362 


0.02206 


8.39264 


0.02470 


8.43899 


0.02748 


24 6 


.29249 


.01961 


.34446 


.02210 


.39344 


.02474 


.43974 


.02753 


28 7 


.29338 


.01965 


.34530 


.02215 


.39423 


.02479 


.44049 


.02757 


32 8 


.29427 


.01969 


.34614 


.02219 


.39502 


.02483 


.44124 


.02762 


36 9 


.29516 


.01973 


.34698 


.02223 


.39581 


.02488 


.44199 


.02767 


40 10 


8.29605 


0.01977 


8.34782 


0.02227 


8.39660 


0.02492 


8.44273 


0.02772 


44 11 


.29694 


.01981 


.34865 


.02232 


.39739 


.02497 


.44348 


.02776 


48 12 


.29783 


.01985 


.34949 


.02236 


.39818 


.02501 


.44423 


.02781 


52 13 


.29872 


.01989 


.35032 


.02240 


.39897 


.02506 


.44498 


.02786 


56 14 


.29960 


.01993 


.35116 


.02245 


.39976 


.02510 


.44572 


.02791 


s ' 


!h 6 m 16 o 


lh gm 17 


Ik 13 18 


Ik 17 m 19 


15 


8.30049 


0.01998 


8.35199 


0.02249 


8.40055 


0.02515 


8.44647 


0.02796 


4 16 


.30137 


.02002 


.35282 


.02253 


.40133 


.02520 


.44721 


.02800 


S 17 


.30226 


.02006 


.35365 


.02258 


.40212 


.02524 


.44796 


.02805 


12 18 


.30314 


.02010 


.35449 


.02262 


.40290 


.02529 


.44870 


.02810 


Jff 19 


.30402 


.02014 


.35532 


.02266 


.40369 


.02533 


.44944 


.02815 


SO 20 


8.30490 


0.02018 


8.35614 


0.02271 


8.40447 


0.02538 


8.45018 


0.02820 


24 21 


.30578 


.02022 


.35697 


.02275 


.40525 


.02542 


.45093 


.02824 


25 22 


.30666 


.02026 


.35780 


.02279 


.40603 


.02547 


.45167 


.02829 


32 23 


.30754 


.02030 


.35863 


.02284 


.40681 


.02552 


.45241 


.02834 


36 24 


.30842 


.02034 


.35945 


.02288 


.40760 


.02556 


.45315 


.02839 


40 25 


8.30929 


0.02038 


8.36028 


0.02292 


8.40837 


0.02561 


8.45388 


0.02844 


44 26 


.31017 


.02043 


.36110 


.02297 


.40915 


.02565 


.45462 


.02849 


4S 27 


.31104 


.02047 


.36193 


.02301 


.40993 


.02570 


.45536 


.02853 


52 28 


.31192 


.02051 


.36275 


.02305 


.41071 . 


.02575 


.45610 


.02858 


Jtf 29 


.31279 


.02055 


.36357 


.02310 


.41149 


.02579 


.45683 


.02863 


s ' 


lh Q 16 


Ik 10 17 


Ik 14 18 


Ik 18 m 19 


30 


8.31366 


0.02059 


8.36439 


0.02314 


8.41226 


0.02584 


8.45757 


0.02868 


4 31 


.31453 


.02063 


.36521 


.02319 


.41304 


.02588 


.45830 


.02873 


8 32 


.31540 


.02067 


.36603 


.02323 


.41381 


.02593 


.45904 


.02878 


i 33 


.31627 


.02071 


.36685 


.02327 


.41459 


.02598 


.45977 


.02883 


iff 34 


.31714 


.02076 


.36767 


.02332 


.41536 


.02602 


.46050 


.02887 


20 35 


8.31800 


0.02080 


8.36849 


0.02336 


8.41613 


0.02607 


8.46124 


0.02892 


#4 36 


.31887 


.02084 


.36930 


.02340 


.41690 


.02612 


.46197 


.02897 


28 37 


.31974 


.02088 


.37012 


.02345 


.41767 


.02616 


.46270 


.02902 


32 38 


.32060 


.02092 


.37093 


.02349 


.41845 


.02621 


.46343 


.02907 


Sff 39 


.32147 


.02096 


.37175 


.02354 


.41921 


.02826 


.46416 


.02912 


40 40 


8.32233 


0.02101 


8.37256 


0.02358 


8.41998 


0.02630 


8.46489 


0.02917 


44 41 


.32319 


.02105 


.37337 


.02363 


.42075 


.02635 


.46562 


. 02922 


48 42 


.32405 


.02109 


.37419 


.02367 


.42152 


.02639 


.46634 


.02926 


52 43 


.32491 


.02113 


.37500 


.02371 


.42229 


.02644 


.46707 


.02931 


5<S 44 


.32577 


.02117 


.37581 


.02376 


.42305 


.02649 


.'-6780 


.02936 


s ' 


Ik 7^ 16 


Ik 11>" 17 


Ik 15 m 18 


Ik 19 19 


45 


8.32663 


0.02121 


8.37662 


0.02380 


8.42382 


0.02653 


8.46852 


0.02941 


4 46 


.32749 


.02126 


.37742 


.02385 


.42458 


.02658 


.46925 


.02946 


S 47 


.32834 


.02130 


.37823 


.02389 


.42535 


.02663 


.46998 


.02951 


J2 48 


.32920 


.02134 


.37904 


.02394 


.42611 


.02668 


.47070 


.02956 


16 49 


.33006 


.02138 


.37985 


.02398 


.42687 


.02672 


.47142 


.02961 


SO 50 


8.33091 


0.02142 


8.38065 


0.02402 


8.42764 


0.02677 


8.47215 


0.02966 


24 51 


.33176 


.02147 


.38146 


.02407 


.42840 


.02682 


.47287 


.02971 


SS 52 


.33262 


.02151 


.38226 


.02411 


.42916 


.02686 


.47359 


.02976 


32 53 


.33347 


.02155 


.38306 


.02416 


.42992 


.02691 


.47431 


.02981 


36 54 


.33432 


.02159 


.38387 


.02420 


.43068 


.02696 


.47503 


.02986 


40 55 


8.33517 


0.02164 


8.38467 


0.02425 


8.43144 


0.02700 


8.47575 


0.02991 


44 56 


.33602 


.02168 


.38547 


.02429 


.43219 


.02705 


.47647 


.02996 


48 57 


.33686 


.02172 


.38627 


.02434 


.43295 


.02710 


.47719 


.03000 


52 58 


.33771 


.02176 


.38707 


.02438 


.43371 


.02715 


.47791 


.03005 


56 59 


.33856 


.02181 


.38787 


.02443 


.43446 


.02719 


.47862 


.03010 


60 60 


8.33940 


0.02185 


8.38867 


0.02447 


8.43522 


0.02724 


8.47934 


0.03015 



Table 10. Haversine Table 



255 



s ' 


l h 20 m 20 


lh 24 21 


lh 28 22 


jh S gm 23 




Hav. 


No. 


Hav. 


No. 


Hav. 


No. 


Hav. 


No. 





8.47934 


0.03015 


8.52127 


0.03321 


8.56120 


0.03641 


8.59931 


0.03975 


4 1 


.48006 


.03020 


.52195 


.03326 


.56185 


.03646 


.59993 


.03980 


8 2 


.48077 


.03025 


.52263 


.03331 


.56250 


.03652 


.60055 


.03986 


12 3 


.48149 


.03030 


.52331 


.03337 


.56315 


.03657 


.60117 


.03992 


16 4 


.48220 


.03035 


.52399 


.03342 


.56379 


.03663 


.60179 


.03998 


20 5 


8.48292 


0.03040 


8.52467 


0.03347 


8.56444 


0.03668 


8.60241 


0.04003 


24 6 


.48363 


.03045 


.52535 


.03352 


.56509 


.03674 


.60303 


.04009 


28 7 


.48434 


.03050 


.52602 


.03358 


.56574 


.03679 


.60365 


.04015 


32 8 


.48505 


.03055 


.52670 


.03363 


.56638 


.03685 


.60426 


.04020 


36 9 


.48576 


.03060 


.52738 


.03368 


.56703 


.03690 


.60488 


.04026 


40 10 


8.48648 


0.03065 


8.52806 


0.03373 


8.56767 


0.03695 


8.60550 


0.04032 


44 11 


.48719 


.03070 


.52873 


.03379 


.56832 


.03701 


.60611 


.04038 


48 12 


.48789 


.03075 


.52941 


.03384 


.56896 


.03706 


.60673 


.04043 


52 13 


.48860 


.03080 


.53008 


.03389 


.56960 


.03712 


.60734 


.04049 


56 14 


.48931 


.03085 


.53076 


.03394 


.57025 


.03717 


.60796 


.04055 


s ' 


lh 21 m 20 


lh 25 m 21 


lh 29 22 


jh Sgm 23 


15 


8.49002 


0.03090 


8.53143 


0.03400 


8.57089 


0.03723 


8.60857 


0.04060 


4 16 


.49073 


.03095 


.53210 


.03405 


.57153 


.03728 


.60919 


.04066 


S 17 


.49143 


.03101 


.53277 


.03410 


.57217 


.03734 


.60980 


.04072 


12 18 


.49214 


.03106 


.53345 


.03415 


.57282 


.03740 


.61041 


.04078 


iff 19 


.49284 


.03111 


.53412 


.03421 


.57346 


.03745 


.61103 


.04083 


20 


8.49355 


0.03116 


8.53479 


0.03426 


8.57410 


0.03751 


8.61164 


0.04089 


24 21 


.49425 


.03121 


.53546 


.03431 


.57474 


.03756 


.61225 


.04095 


2S 22 


.49496 


.03126 


.53613 


.03437 


.57538 


.03762 


.61286 


.04101 


32 23 


.49566 


.03131 


.53680 


.03442 


.57601 


.03767 


.61347 


.04106 


30 24 


.49636 


.03136 


.53747 


.03447 


.57665 


.03773 


.61408 


.04112 


40 25 


8.49706 


0.03141 


8.53814 


0.03453 


8.57729 


0.03778 


8.61469 


0.04118 


44 26 


.49777 


.03146 


.53880 


.03458 


.57793 


.03784 


.61530 


.04124 


45 27 


.49847 


.03151 


.53947 


.03463 


.57856 


.03789 


.61591 


.04130 


52 28 


.49917 


.03156 


.54014 


.03468 


.57920 


.03795 


.61652 


.04135 


56 29 


.49987 


.03161 


.54080 


.03474 


.57984 


.03800 


.61713 


.04141 


s ' 


lh 22 m 20 


lh gffi* 21 


lh so m 22 


/* 34 m 23 


30 


8.50056 


0.03166 


8.54147 


0.03479 


8.58047 


0.03806 


8.61773 


0.04147 


^ 31 


.50126 


.03171 


.54214 


.03484 


.58111 


.03812 


.61834 


.04153 


8 32 


.50196 


.03177 


.54280 


.03490 


.58174 


.03817 


.61895 


.04159 


10 33 


.50266 


.03182 


.54346 


.03495 


.58238 


.03823 


.61955 


.04164 


/ff 34 


.50335 


.03187 


.54413 


.03500 


.58301 


.03828 


.62016 


.04170 


20 35 


8.50405 


0.03192 


8.54479 


0.03506 


8.58364 


0.03834 


8.62077 


0.04176 


24 36 


.50475 


.03197 


.54545 


.03511 


.58427 


.03839 


.62137 


.04182 


28 37 


.50544 


.03202 


.54612 


.03517 


.58491 


.03845 


.62197 


.04188 


32 38 


.50614 


.03207 


.54678 


.03522 


.58554 


.03851 


.62258 


.04194 


36 39 


.50683 


.03212 


.54744 


.03527 


.58617 


.03856 


.62318 


.04199 


40 40 


8.50752 


0.03218 


8.54810 


0.03533 


8.58680' 


0.03862 


8.62379 


0.04205 


44 41 


.50821 


.03223 


.54876 


.03538 


.58743 


.03867 


.62439 


.04211 


48 42 


.50891 


.03228 


.54942 


.03543 


.58806 


.03873 


.62499 


.04217 


52 43 


.50960 


.03233 


.55008 


.03549 


.58869 


.03879 


.62559 


.04223 


5ff 44 


.51029 


.03238 


.55073 


.03554 


.58932 


.03884 


.62619 


.04229 


s ' 


lh 23 20 


lh 2 7 m 21 


lh Sim 22 


lh 3 5 m 23 


45 


8.51098 


0.03243 


8.55139 


0.03560 


8.58994 


0.03890 


8.62680 


0.04234 


4 46 


.51167 


.03248 


.55205 


.03565 


.59057 


.03896 


.62740 


.04240 


S 47 


.51236 


.03254 


.55271 


.03570 


.59120 


.03901 


.62800 


.04246 


J2 48 


.51305 


.03259 


.55336 


.03576 


.59183 


.03907 


.62860 


.04252 


16 49 


.51374 


.03264 


.55402 


.03581 


.59245 


.03912 


.62919 


.04258 


20 50 


8.51442 


0.03269 


8.55467 


0.03587 


8.59308 


0.03918 


8.62979 


0.04264 


24 51 


.51511 


.03274 


.55533 


.03592 


.59370 


.03924 


.63039 


.04270 


25 52 


.51580 


.03279 


.55598 


.03597 


.59433 


.03929 


.63099 


.04276 


32 53 


.51648 


.03285 


.55664 


.03603 


.59495 


.03935 


.63159 


.04281 


Sff 54 


.51717 


.03290 


.55729 


.03608 


.59558 


.03941 


.63218 


.04287 


40 55 


8.51785 


0.03295 


8.55794 


0.03614 


8.59620 


0.03946 


8.63278 


0.04293 


44 56 


.51854 


.03300 


.55859 


.03619 


.59682 


.03952 


.63338 


.04299 


4S 57 


.51922 


.03305 


.55925 


.03624 


.59745 


.03958 


.63397 


.04305 


.52 58 


.51990 


.03311 


.55990 


.03630 


.59807 


.03963 


.63457 


.04311 


56 59 


.52058 


.03316 


.56055 


.03635 


.59869 


.03969 


.63516 


.04317 


60 60 


8.52127 


0.03321 


8.56120 


0.03641 


8.59931 


0.03975 


8.63576 


0.04323 



256 



Table 10. Haversine Table 



s ' 


lh Sffn 24 


1>> 40 m 25 


I* 44 m 26 


lh .> 27 




Hav. 


No. 


Hav. 


No. 


Hav. 


No. 


Hav. 


No. 





8.63576 


0.04323 


8.67067 


0.04685 


8.70418 


0.05060 


8.73637 


0.05450 


4 1 


.63635 


.04329 


.67124 


.04691 


.70472 


.05067 


.73690 


.05456 


8 2 


.63695 


.04335 


.67181 


.04697 


.70527 


.05073 


.73742 


.05463 


12 3 


.63754 


.04340 


.67238 


.04703 


.70582 


.05079 


.73795 


.05470 


16 4 


.63813 


.04346 


.67295 


.04709 


.70636 


.05086 


.73847 


.05476 


20 5 


8.63872 


0.04352 


8.67352 


0.04715 


8.70691" 


0.05092 


8.73900 


0.05483 


24 6 


.63932 


.04358 


.67409 


.04722 


.70745 


.05099 


.73952 


.05489 


28 7 


-.63991 


.04364 


.67465 


.04728 


.70800 


.05105 


.74005 


.05496 


32 8 


.64050 


.04370 


.67522 


.04734 


.70854 


.05111 


.74057 


.05503 


36 9 


.64109 


.04376 


.67579 


.04740 


.70909 


.05118 


.74109 


.05509 


40 10 


8.64168 


0.04382 


8.67635 


0.04746 


8.70963 


0.05124 


8.74162 


0.05516 


44 11 


.64227 


.04388 


.67692 


.04752 


.71017 


.05131 


.74214 


.05523 


48 12 


.64286 


.04394 


.67748 


.04759 


.71072 


.05137 


.74266 


.05529 


52 13 


.64345 


.04400 


.67805 


.04765 


.71126 


.05144 


.74318 


.05536 


56 14 


.64404 


.04405 


.67861 


.04771 


.71180 


.05150 


.74371 


.05542 


s ' 


lh 3 jm 24 


lh um 25 o 


1* 45 m 26 


lh ^y 27 


15 


8.64463 


0.04412 


8.67918 


0.04777 


8.71234 


0.05156 


8.74423 


0.05549 


', 16 


.64521 


.04418 


.67974 


.04783 


.71289 


.05163 


.74475 


.05556 


S 17 


.64580 


.04424 


.68030 


.04790 


.71343 


.05169 


.74527 


.05562 


12 18 


.64639 


.04430 


.68087 


.04796 


.71397 


.05176 


.74579 


.05569 


J6 19 


.64697 


.04436 


.68143 


.04802 


.71451 


.05182 


.74631 


.05576 


20 20 


8.64756 


0.04442 


8.68199 


0.04808 


8.71505 


0.05189 


8.74683 


0.05582 


24 21 


.64815 


.04448 


.68256 


.04815 


.71559 


.05195 


.74735 


.05589 


2S 22 


.64873 


.04454 


.68312 


.04821 


.71613 


.05201 


.74787 


.05596 


32 23 


.64932 


.04460 


.68368 


.04827 


.71667 


.05208 


.74839 


.05603 


36 24 


.64990 


.04466 


.68424 


.04833 


.71721 


.05214 


.74890 


.05609 


40 25 


8.65049 


0.04472 


8.68480 


0.04839 


8.71774 


0.05221 


8.74942 


0.05616 


44 26 


.65107 


.04478 


.68536 


.04846 


.71828 


.05227 


.74994 


.05623 


48 27 


.65165 


.04484 


.68592 


.04852 


.71882 


.05234 


.75046 


.05629 


52 28 


.65224 


.04490 


.68648 


.04858 


.71936 


.05240 


.75097 


.05636 


56 29 


.65282 


.04496 


.68704 


.04864 


.71989 


.05247 


.75149 


.05643 


s ' 


lh 28 m 24 


lh .> 25 


1* 4& 26 


1* 50 27 


30 


8.65340 


0.04502 


8.68760 


0.04871 


8.72043 


0.05253 


8.75201 


0.05649 


4 31 


.65398 


.04508 


.68815 


.04877 


.72097 


.05260 


.75252 


.05656 


8 32 


.65456 


.04514 


.68871 


.04883 


.72150 


.05266 


.75304 


.05663 


.72 33 


.65514 


.04520 


.68927 


.04890 


.72204 


.05273 


.75355 


.05670 


^6 34 


.65572 


.04526 


.68983 


.048% 


.72257 


.05279 


.75407 


.05676 


20 35 


8.65630 


0.04532 


8.69038 


0.04902 


8.72311 


0.05286 


8.75458' 


0.05683 


24 36 


.65688 


.04538 


.69094 


.04908 


.72364 


.05292 


.75510 


.05690 


28 37 


.65746 


.04544 


.69149 


.04915 


.72418 


.05299 


.75561 


.05697 


32 38 


.65804 


.04550 


.69205 


.04921 


.72471 


.05305 


.75613 


.05703 


36 39 


.65862 


.04556 


.69260 


.04927 


.72525 


.05312 


.75664 


.05710 


40 40 


8.65920 


0.04562 


8.69316 


0.04934 


8.72578 


0.05318 


8.75715 


0.05717 


44 41 


.65978 


.04569 


.69371 


.04940 


.72631 


.05325 


.75767 


.05724 


48 42 


.66035 


.04575 


.69427 


.04946 


.72684 


.05331 


.75818 


.05730 


52 43 


.66093 


.04581 


.69482 


.04952 


.72738 


.05338 


.75869 


.05737 


56 44 


.66151 


.04587 


.69537 


.04959 


.72791 


.05345 


:, 5920 


.05744 


s ' 


lh gym 24 


lh 43 25 


Jh 47 m 26 


lh 51*. 27 


45 


8.66208 


0.04593 


8.69593 


0.04965 


8.72844 


0.05351 


8.75972 


O.C5751 


4 46 


.66266 


.04599 


.69648 


.04971 


.72897 


.05358 


.76023 


.05757 


S 47 


.66323 


.04605 


.69703 


.04978 


.72950 


.05364 


.76074 


.C5764 


.72 48 


.66381 


.04611 


.69758 


.04984 


.73003 


.05371 


.76125 


.05771 


16 49 


.66438 


.04617 


.69814 


.04990 


.73056 


.05377 


.76176 


.05778 


20 50 


8.66496 


0.04623 


8.69869 


0.04997 


8.73109 


0.05384 


8.76227 


O.C5785 


24 51 


.66553 


.04629 


.69924 


.05003 


.73162 


.05390 


.76278 


.05791 


28 52 


.66610 


.04636 


.69979 


.05009 


.73215 


.05397 


.76329 


.05798 


32 53 


.66668 


.04642 


.70034 


.05016 


.73268 


.05404 


.76380 


.05805 


36 54 


.66725 


.04648 


.70089 


.05022 


.73321 


.05410 


.76431 


.05812 


40 55 


8.66782 


0.04654 


8.70144 


0.05028 


8.73374 


0.05417 


8.76481 


0.05819 


44 56 


.66839 


.04660 


.70198 


.05035 


.73426 


.05423 


.76532 


.05825 


48 57 


.66896 


.04666 


.70253 


.05041 


.73479 


.05430 


.76583 


.05832 


52 58 


.66953 


.04672 


.70308 


.05048 


.73532 


.05436 


.76634 


.05839 


56 59 


.67010 


.04678 


.70363 


.05054 


.73584 


.05443 


.76684 


.05846 


60 60 


8.67067 


0.04685 


8.70418 


0.05060 


8.73637 


0.05450 


8.76735 


0.05853 



Table 10. Haversine Table 



257 



s ' 


7* 52 m 28 


J* 56'" 29 


2 h o m 30 


2* 4 m 31 




Hav. 


No. 


Hav. No. 


Hav. 


No. 


Hav. 


No. 





8.76735 


0.05853 


8.79720 


0.06269 


8.82599 


0.06699 


8.85380 


0.07142 


4 1 


.76786 


.05859 


.79769 


.06276 


.82646 


.06706 


.85425 


.07149 


8 2 


.76836 


.05866 


.79818 


.06283 


.82694 


.06713 


.85471 


.07157 


12 3 


.76887 


.05873 


.79866 


.06290 


.82741 


.06721 


.85516 


.07164 


16 4 


.76938 


.05880 


.79915 


.06297 


.82788 


.06728 


.85562 


.07172 


20 5 


8.76988 


0.05887 


8.79964 


0.06304 


8.82835 


0.06735 


8.85607 


0.07179 


24 6 


.77039 


.05894 


.80013 


.06311 


.82882 


.06742 


.85653 


.07187 


28 7 


.77089 


.05901 


.80061 


.06318 


.82929 


.06750 


.85698 


.07194 


32 8 


.77139 


.05907 


.80110 


.06326 


.82976 


.06757 


.85743 


.07202 


36 9 


.77190 


.05914 


.80158 


.06333 


.83023 


.06764 


.85789 


.07209 


40 10 


8.77240 


0.05921 


8.80207 


0.06340 


8.83069 


0.06772 


8.85834 


0.07217 


44 11 


.77291 


.05928 


.80256 


.06347 


.83116 


.06779 


.85879 


.07224 


48 12 


.77341 


.05935 


.80304 


.06354 


.83163 


.06786 


.85925 


.07232 


52 13 


.77391 


.05942 


.80353 


.06361 


.83210 


.06794 


.85970 


.07239 


56 14 


.77441 


.05949 


.80401 


.06368 


.83257 


.06801 


.86015 


.07247 


s ' 


7* 53 m 28 


lh j7> 29 


2* l m 30 


2 h 5 m 31 


15 


8.77492 


0.05955 


8.80449 


0.06375 


8.83303 


0.06808 


8.86060 


0.07254 


4 16 


.77542 


.05982 


.80498 


.06382 


.83350 


.06816 


.86105 


.07262 


S 17 


.77592 


.05969 


.80546 


.06389 


.83397 


.06823 


.86151 


.07270 


12 18 


.77642 


.05976 


.80595 


.06397 


.83444 


.06830 


.86196 


.07277 


J6 19 


.77692 


.05983 


.80643 


.06404 


.83490 


.06838 


.86241 


.07285 


20 20 


8.77742 


0.05990 


8.80691 


0.06411 


8.83537 


0.06845 


8.86286 


0.07292 


24 21 


.77792 


.05997 


.80739 


.06418 


.83583 


.06852 


.86331 


.07300 


2S 22 


.77842 


.06004 


.80788 


.06425 


.83630 


.06860 


.86376 


.07307 


32 23 


.77892 


.06011 


.80836 


.06432 


.83676 


.06867 


.86421 


.07315 


36 24 


.77942 


.06018 


.80884 


.06439 


.83723 


.06874 


.86466 


.07322 


40 25 


8.77992 


0.06024 


8.80932 


0.06446 


8.83769 


0.06882 


8.86511 


0.07330 


44 26 


.78042 


.06031 


.80980 


.06454 


.83816 


.06889 


.86556 


.07338 


45 27 


.78092 


.06038 


.81028 


.06461 


.83862 


.06896 


.86600 


.07345 


52 28 


.78142 


.06045 


.81076 


.06468 


.83909 


.06904 


.86645 


.07353 


56 29 


.78191 


.06052 


.81124 


.06475 


.83955 


.06911 


.86690 


.07360 


s ' 


1* 54 m 28 


lh o8 m 29 


gh gm 3Q 


2 h Qm 31 


30 


8.78241 


0.06059 


8.81172 


0.06482 


8.84002 


0.06919 


8.86735 


0.07368 


4 31 


.78291 


.06066 


.81220 


.06489 


.84048 


.06926 


.86780 


.07376 


8 32 


.78341 


.06073 


.81268 


.06497 


.84094 


.06933 


.86825 


.07383 


72 33 


.78390 


.06080 


.81316 


.06504 


.84140 


.06941 


.86869 


.07391 


16 34 


.78440 


.06087 


.81364 


.06511 


.84187 


.06948 


.86914 


.07398 


20 35 


8.78490 


0.06094 


8.81412 


0.06518 


8.84233 


0.06956 


8.86959 


0.07406 


24 36 


.78539 


.06101 


.81460 


.06525 


.84279 


.06963 


.87003 


.07414 


28 37 


.78589 


.06108 


.81508 


.06532 


.84325 


.06970 


.87048 


.07421 


32 38 


.78638 


.06115 


.81555 


.06540 


.84371 


.06978 


.87093 


.07429 


36 39 


.78688 


.06122 


.81603 


.06547 


.84417 


.06985 


.87137 


.07437 


40 40 


8.78737 


0.06129 


8.81651 


0.06554 


8.84464 


0.06993 


8.87182 


0.07444 


44 41 


.78787 


.06136 


.81699 


.06561 


.84510 


.07000 


.87226 


.07452 


48 42 


.78836 


.06143 


.81746 


.06568 


.84556 


.07007 


.87271 


.07459 


52 43 


.78885 


.06150 


.81794 


.06576 


.84602 


.07015 


.87315 


.07467 


56 44 


.78935 


.06157 


.81841 


.06583 


.84648 


07022 


.87360 


.07475 


s . ' 


Ik 55 m 28 


lh 59* 29 


2 h 3" 1 30 


2 h 7 31 


45 


8.78984 


0.06164 


8.81889 


0.06590 


8.84694 


0.07030 


8.87404 


0.07482 


4 46 


.79033 


.06171 


.81937 


.06597 


.84740 


.07037 


.87448 


.07490 


S 47 


.79082 


.06178 


.81984 


.06605 


.84785 


.07045 


.87493 


.07498 


12 48 


.79132 


.06185 


.82032 


.06612 


.84831 


.07052 


.87537 


.07505 


16 49 


.79181 


.06192 


.82079 


.06619 


.84877 


.07059 


.87582 


.07513 


20 50 


8.79230 


0.06199 


8.82126 


0.06626 


8.84923 


0.07067 


8.87626 


0.07521 


24 51 


.79279 


.06206 


.82174 


.06633 


.84969 


.07074 


.87670 


.07528 


28 52 


.79328 


.06213 


.82221 


.06641 


.85015 


.07082 


.87714 


.07536 


32 63 


.79377 


.06220 


.82269 . 


.06648 


.85060 


.07089 


.87759 


.07544 


36 54 


.79426 


.06227 


.82316 


.06655 


.85106 


.07097 


.87803 


.07551 


40 65 


8.79475 


0.06234 


8.82363 


0.06662 


8.85152 


0.07104 


8.87847 


0.07559 


44 56 


.79524 


.06241 


.82410 


.06670 


.85197 


.07112 


.87891 


.07567 


48 57 


.79573 


.06248 


.82458 


.06677 


.85243 


.07119 


.87935 


.07574 


5J 58 


.79622 


.06255 


.82505 


.06684 


.85289 


.07127 


.87980 


.07582 


56 59 


.79671 


.06262 


.82552 


.06691 


.85334 


.07134 


.88024 


.07590 


60 60 


8.79720 


0.06269 


8.82599 


0.06699 


8.85380 


0.07142 


8.88068 


0.07598 



258 



Table 10. Haversine Table 



s ' 


%h g 32 


2 h 12 33 


2 h 16 m 34 


2* 20 m 35 




HOT. 


No. 


Hav. 


No. 


Hav. 


No. 


Hav. 


No. 





8.88068 


0.07598 


8.90668 


0.08066 


8.93187 


0.08548 


8.95628 


0.09042 


4 1 


.88112 


.07605 


.90711 


.08074 


.93228 


.08556 


.95668 


.09051 


8 2 


.88156 


.07613 


.90754 


.08082 


.93270 


.08564 


.95709 


.09059 


12 3 


.88200 


.07621 


.90796 


.08090 


.93311 


.08573 


.95749 


.09067 


18 4 


.88244 


.07628 


.90839 


.08098 


.93352 


.08581 


.95789 


.09076 


20 5 


8.88288 


0.07636 


8.90881 


0.08106 


8.93393 


0.08589 


8.95828 


0.09084 


24 6 


.88332 


.07644 


.90924 


.08114 


.93435 


.08597 


.95868 


.09093 


28 7 


.88375 


.07652 


.90966 


.08122 


.93476 


.08605 


.95908 


.09101 


32 8 


.88419 


.07659 


.91009 


.08130 


.93517 


.08613 


.95948 


.09109 


36 9 


.88463 


.07667 


.91051 


.08138 


.93558 


.08621 


.95988 


.09118 


40 10 


8.88507 


0.07675 


8.91094 


0.08146 


8.93599 


0.08630 


8.96028 


0.09126 


44 11 


.88551 


.07683 


.91136 


.08154 


.93640 


.08638 


.96068 


.09134 


48 12 


.88595 


.07690 


.91179 


.08162 


.93681 


.08646 


.96108 


.09143 


52 13 


.88638 


.07698 


.91221 


.08170 


.93722 


.08654 


.96148 


.09151 


56 14 


.88682 


.07706 


.91263 


.08178 


.93764 


.08662 


.96187 


.09160 


s ' 


2 h Sf" 1 32 


2 h IS" 1 33 


2 h 17 m 34 


%h 21 m 35 


15 


8.88726 


0.07714 


8.91306 


0.08186 


8.93805 


0.08671 


8.96227 


0.09168 


4 16 


.88769 


.07721 


.91348 


.08194 


.93846 


.08679 


.96267 


.09176 


5 17 


.88813 


.07729 


.91390 


.08202 


.93886 


.08687 


.96307 


.09185 


12 18 


.88857 


.07737 


.91432 


.08210 


.93927 


.08695 


.96346 


.09193 


iff 19 


.88900 


.07745 


.91475 


.08218 


.93968 


.08703 


.96386 


.09202 


20 


8.88944 


0.07752 


8.91517 


0.08226 


8.94009 


0.08711 


8.96426 


0.09210 


24 21 


.88988 


.07760 


.91559 


.08234 


.94050 


.08720 


.96465 


.09218 


25 22 


.89031 


.07768 


.91601 


.08242 


.94091 


.08728 


.96505 


.09227 


32 23 


.89075 


.07776 


.91643 


.08250 


.94132 


.08736 


.96545 


.09235 


35 24 


.89118 


.07784 


.91685 


.08258 


.94173 


.08744 


.96584 


.09244 


40 25 


8.89162 


0.07791 


8.91728 


0.08266 


8.94213 


0.08753 


8.96624 


0.09252 


44 26 


.89205 


.07799 


.91770 


.08274 


.94254 


.08761 


.96663 


.09260 


45 27 


.89248 


.07807 


.91812 


.08282 


.94295 


.08769 


.96703 


.09269 


52 28 


.89292 


.07815 


.91854 


.08290 


.94336 


.08777 


.96742 


.09277 


56 29 


.89335 


.07823 


.91896 


.08298 


.94376 


.08785 


.96782 


.09286 


s ' 


gh w m 32 


2 h 14 m 33 


2?A 18 m 34 


%h 22" 1 35 


30 


8.89379 


0.07830 


8.91938 


0.08306 


8.94417 


0.08794 


8.96821 


0.09294 


4 31 


.89422 


.07838 


.91980 


.08314 


.94458 


.08802 


.96861 


.09303 


8 32 


.89465 


.07846 


.92022 


.08322 


.94498 


.08810 


.96900 


.09311 


12 33 


.89509 


.07854 


.92064 


.08330 


.94539 


.08818 


.96940 


.09320 


15 34 


.89552 


.07862 


.92105 


.08338 


.94580 


.08827 


.96979 


.09328 


20 35 


8.89595 


0.07870 


8.92147 


0.08346 


8.94620 


0.08835 


8.97018 


0.09337 


24 36 


.89638 


.07877 


.92189 


.08354 


.94661 


.08843 


.97058 


.09345 


28 37 


.89681 


.07885 


.92231 


.08362 


.94701 


.08851 


.97097 


.09353 


32 38 


.89725 


.07893 


.92273 


.08370 


.94742 


.08860 


.97136 


.09362 


35 39 


.89768 


.07901 


.92315 


.08378 


.94782 


.08868 


.97176 


.09370 


40 40 


8.89811 


0.07909 


8.92356 


0.08386 


8.94823 


0.08876 


8.97215 


0.09379 


44 41 


.89854 


.07917 


.92398 


.08394 


.94863 


.08885 


.97254 


.09387 


48 42 


.89897 


.07924 


.92440 


.08402 


.94904 


.08893 


.97294 


.09396 


52 43 


.89940 


.07932 


.92482 


.08410 


.94944 


.08901 


.97333 


.09404 


55 44 


.89983 


.07940 


.92523 


.08418 


.94985 


.08909 


97372 


.09413 


s ' 


2 h ll m 32 


2k 15 m 33 


2 h 19 m 34 


2 h 23 35 


45 


8.90026 


0.07948 


8.92565 


0.08427 


8.95025 


0.08918 


8.97411 


0.09421 


4 46 


.90069 


.07956 


.92607 


.08435 


.95065 


.08926 


.97450 


.09430 


S 47 


.90112 


.07964 


.92648 


.08443 


.95106 


.08934 


.97489 


.09438 


12 48 


.90155 


.07972 


.92690 


.08451 


.95146 


.08943 


.97529 


.09447 


16 49 


.90198 


.07980 


.92731 


.08459 


.95186 


.08951 


.97568 


.09455 


20 50 


8.90241 


0.07987 


8.92773 


0.08467 


8.95227 


0.08959 


8.97607 


0.09464 


24 51 


.90284 


.07995 


.92814 


.08475 


.95267 


.08967 


.97646 


.09472 


28 52 


.90326 


.08003 


.92856 


.08483 


.95307 


.08976 


.97685 


.09481 


32 53 


.90369 


.08011 


.92897 


.08491 


.95347 


.08984 


.97724 


.09489 


35 54 


.90412 


.08019 


.92939 


.08499 


.95388 


.08992 


.97763 


.09498 


40 55 


8.90455 


0.08027 


8.92980 


0.08508 


8.95428 


0.09001 


8.97802 


0.09506 


44 56 


.90498 


.08035 


.93022 


.08516 


.95468 


.09009 


.97841 


.09515 


48 57 


.90540 


.08043 


.93063 


.08524 


.95508 


.09017 


.97880 


.09524 


52 58 


.90583 


.08051 


.93104 


.08532 


.95548 


.09026 


.97919 


.09532 


55 59 


.90626 


.08059 


.93146 


.08540 


.95588 


.09034 


.97958 


.09541 


60 60 


8.90668 


0.08066 


8.93187 


0.08548 


8.95628 


0.09042 


8.97997 


0.09549 



Table 10. Harersine Table 



259 



s ' 


2* 24 m 36 


2 h 28 37 


2 h 32 m 38 


2 h 36 m 39 




Bar. 


No. 


Bav. 


No. 


Bar. 


No. 


Bav. 


No. 





8.97997 


0.09549 


9.00295 


0.10068 


9.02528 


0.10599 


9.04699 


0.11143 


4 1 


.98035 


.09558 


.00333 


.10077 


.02565 


.10608 


.04735 


.11152 


8 2 


.98074 


.09566 


.00371 


.10086 


.02602 


.10617 


.04770 


.11161 


12 3 


.98113 


.09575 


.00408 


.10095 


.02638 


.10626 


.04806 


.11170 


16 4 


.98152 


.09583 


.00446 


.10103 


.02675 


.10635 


.04842 


.11179 


20 5 


8.98191 


0.09592 


9.00484 


0.10112 


9.02712 


0.10644 


9.04877 


0.11189 


24 6 


.98229 


.09601 


.00522 


.10121 


.02748 


.10653 


.04913 


.11198 


28 7 


.98268 


.09609 


.00559 


.10130 


.02785 


.10662 


.04948 


.11207 


32 8 


.98307 


.09618 


.00597 


.10138 


.02821 


.10671 


.04984 


.11216 


36 9 


.98346 


.09626 


.00634 


.10147 


.02858 


.10680 


.05019 


.11225 


40 10 


8.98384 


0.09635 


9.00672 


0.10156 


9.02894 


0.10689 


9.05055 


0.11234 


44 11 


.98423 


.09643 


.00710 


.10165 


.02931 


.10698 


.05090 


.11244 


48 12 


.98462 


.09652 


.00747 


.10174 


.02967 


.10707 


.05126 


.11253 


52 13 


.98500 


.09661 


.00785 


.10182 


.03004 


.10716 


.05161 


.11262 


56 14 


.98539 


.09669 


.00822 


.10191 


.03040 


.10725 


.05197 


.11271 


s ' 


2* 25 m 36 


2 h 29 m 37 


2 h 33 m 38 


2* 37 m 39 


15 


8.98578 


0.09678 


9.00860 


0.10200 


9.03077 


0.10734 


9.05232 


0.11280 


4 16 


.98616 


.09686 


.00897 


.10209 


.03113 


.10743 


.05268 


.11290 


S 17 


.98655 


.09695 


.00935 


.10218 


.03150 


.10752 


.05303 


.11299 


12 18 


.98693 


.09704 


.00972 


.10226 


.03186 


.10761 


.05339 


.11308 


7<5 19 


.98732 


.09712 


.01009 


.10235 


.03222 


.10770 


.05374 


.11317 


20 


8.98770 


0.09721 


9.01047 


0.10244 


9.03259 


0.10779 


9.05409 


0.11326 


24 21 


.98809 


.09729 


.0-1084 


.10253 


.03295 


.10788 


.05445 


.11336 


25 22 


.'.ISM7 


.09738 


.01122 


.10262 


.03331 


.10797 


.05480 


.11345 


32 23 


.DSSS6 


.09747 


.01159 


.10270 


.03368 


.10806 


.05515 


.11354 


36 24 


.98924 


.09755 


.01196 


.10279 


.03404 


.10815 


.05551 


.11363 


40 25 


8.98963 


0.09764 


9.01234 


0.10288 


9.03440 


0.10824 


9.05586 


0.11373 


44 26 


.99001 


.09773 


.01271 


.10297 


.03476 


.10833 


.05621 


.11382 


4S 27 


.99039 


.09781 


.01308 


.10306 


.03513 


.10842 


.05656 


.11391 


52 28 


.99078 


.09790 


.01345 


.10315 


.03549 


.10851 


.05692 


.11400 


56 29 


.99116 


.09799 


.01383 


.10323 


.03585 


.10861 


.05727 


.11410 


s ' 


2* 26 1 " 36 


2* SO 37 


2 h 34 m 38 


2* 38 39 


30 


8.99154 


0.09807 


9.01420 


0.10332 


9.03621 


0.10870 


9.05762 


0.11419 


4 31 


.99193 


.09816 


.01457 


.10341 


.03657 


.10879 


.05797 


.11428 


8 32 


.99231 


.09824 


.01494 


.10350 


.03694 


.10888 


.05832 


.11437 


.72 33 


.99269 


.09833 


.01531 


.10359 


.03730 


.10897 


.05867 


.11447 


16 34 


.99307 


.09842 


.01569 


.10368 


.03766 


.10906 


.05903 


.11456 


20 35 


8.99346 


0.09850 


9.01606 


0.10377 


9.03802 


0.10915 


9.05938 


0.11465 


24 36 


.99384 


.09859 


.01643 


.10386 


.03838 


.10924 


.05973 


.11474 


28 37 


.99422 


.09868 


.01680 


.10394 


.03874 


.10933 


.06008 


.11484 


32 38 


.99460 


.09876 


.01717 


.10403 


.03910 


.10942 


.06043 


.11493 


36 39 


.99498 


.09885 


.01754 


.10412 


.03946 


.10951 


.06078 


.11502 


40 40 


8.99536 


0.09894 


9.01791 


0.10421 


9.03982 


0.10960 


9.06113 


0.11511 


44 41 


.99575 


.09903 


.01828 


.10430 


.04018 


.10969 


.06148 


.11521 


48 42 


.99613 


.09911 


.01865 


.10439 


.04054 


.10978 


.06183 


.11530 


52 43 


.99651 


.09920 


.01902 


.10448 


.04090 


.10988 


.06218 


.11539 


56 44 


.99689 


.09929 


.01939 


.10457 


.04126 


.10997 


.0(1253 


.11549 


8 ' 


2* 27 m 36 


2 h 31 m 37 


2* 35 m 38 


2* 39 m 39 


45 


8.99727 


0.09937 


9.01976 


0.10466 


9.04162 


0.11006 


9.06288 


0.11558 


4 46 


.99765 


.09946 


.02013 


.10474 


.04198 


.11015 


.06323 


.11567 


S 47 


.99803 


.09955 


.02050 


.10483 


.04234 


.11024 


.06358 


.11577 


.72 48 


.99841 


.09963 


.02087 


.10492 


.04270 


.11033 


.06393 


.11586 


16 49 


.99879 


.09972 


.02124 


.10501 


.04306 


.11042 


.06428 


.11595 


20 50 


8.99917 


0.09981 


9.02161 


0.10510 


9.04341 


0.11051 


9.06462 


0.11604 


24 51 


.99955 


.09990 


.02197 


.10519 


.04377 


.11060 


.06497 


.11614 


28 52 


.99993 


.09998 


.02234 


.10528 


.04413 


.11070 


.06532 


.11623 


32 53 


9.00031 


.10007 


.02271 


.10637 


.04449 


.11079 


.06567 


.11632 


36 54 


.00068 


.10016 


.02308 


.10546 


.04485 


.11088 


.06602 


.11642 


40 55 


9.00106 


0.10025 


9.02345 


0.10555 


9.04520 


0.11097 


9.06637 


0.11651 


44 56 


.00144 


.10033 


.02381 


.10564 


.04556 


.11106 


.06871 


.11660 


48 57 


.00182 


.10042 


.02418 


.10573 


.04592 


.11115 


.06706 


.11670 


52 58 


.00220 


.10051 


.02455 


.10582 


.04628 


.11124 


.06741 


.11679 


56 59 


.00258 


.10059 


.02492 


.10591 


.04663 


.11134 


.06776 


.11688 


60 60 


9.00295 


010068 


9.02528 


0.10599 


9.04699 


0.11143 


9.06810 


0.11698 



260 



Table 10. Haversine Table 



s ' 


2 h 40 m 40 


2* 44 m 41 


2 h 48 42 


2* 52" 43 




Hav. 


No. 


Hav. 


No. 


Hav. 


No. 


Hav. 


No. 





9.06810 


0.11698 


9.08865 


0.12265 


9.10866 


0.12843 


9.12815 


0.13432 


4 1 


.06845 


.11707 


.08899 


.12274 


.10899 


.12852 


.12847 


.13442 


5 2 


.06880 


.11716 


.08933 


.12284 


.10932 


.12862 


.12879 


.13452 


12 3 


.06914 


.11726 


.08966 


.12293 


.10965 


.12872 


.12911 


.13462 


16 4 


.06949 


.11735 


.09000 


.12303 


.10997 


.12882 


.12943 


.13472 


20 5 


9.06984 


0.11745 


9.09034 


0.12312 


9.11030 


0.12891 


9.12975 


0.13482 


24 6 


.07018 


.11754 


.09068 


.12322 


.11063 


.12901 


.13007 


.13492 


28 7 


.07053 


.11763 


.09101 


.12331 


.11096 


.12911 


.13039 


.13502 


32 8 


.07088 


.11773 


.09135 


.12341 


.11129 


.12921 


.13071 


.13512 


'36 9 


.07122 


.11782 


.09169 


.12351 


.11161 


.12930 


.13103 


.13522 


40 10 


9.07157 


0.11791 


9.09202 


0.12360 


9.11194 


0.12940 


9.13135 


0.13532 


44 11 


.07191 


.11801 


.09236 


.12370 


.11227 


.12950 


.13167 


.13542 


48 12 


.07226 


.11810 


.09269 


.12379 


.11260 


.12960 


.13199 


.13552 


52 13 


.07260 


.11820 


.09303 


.12389 


.11292 


.12970 


.13231 


.13562 


56 14 


.07295 


.11829 


.09337 


.12398 


.11325 


.12979 


.13263 


.13571 


s ' 


2 h 41 m 40 


2* 45 TO 41 


2 h 49 m 42 


2* 53" 1 43 


15 


9.07329 


0.11838 


9.09370 


0.12408 


9.11358 


0.12989 


9.13295 


0.13581 


4 16 


.07364 


.11848 


.09404 


.12418 


.11391 


.12999 


.13326 


.13591 


S 17 


.07398 


.11857 


.09437 


.12427 


.11423 


.13009 


.13358 


.13601 


12 18 


.07433 


.11867 


.09471 


.12437 


.11456 


.13018 


.13390 


.13611 


.70 19 


.07467 


.11876 


.09504 


.12446 


.11489 


.13028 


.13422 


.13621 


20 20 


9.07501 


0.11885 


9.09538 


0.12456 


9.11521 


0.13038 


9.13454 


0.13631 


24 21 


.07536 


.11895 


.09571 


.12466 


.11554 


.13048 


.13486 


.13641 


2S 22 


.07570 


.11904 


.09605 


.12475 


.11586 


.13058 


.13517 


.13651 


32 23 


.07605 


.11914 


.09638 


.12485 


.11619 


.13067 


.13549 


.13661 


36 24 


.07639 


.11923 


.09672 


.12494 


.11652 


.13077 


.13581 


.13671 


40 25 


9.07673 


0.11933 


9.09705 


0.12504 


9.11684 


0.13087 


9.13613 


0.13681 


44 26 


.07708 


.11942 


.09739 


.12514 


.11717 


.13097 


.13644 


.13691 


48 27 


.07742 


.11951 


.09772 


.12523 


.11749 


.13107 


.13676 


.13701 


52 28 


.07776 


.11961 


.09805 


.12533 


.11782 


.13116 


.13708 


.13711 


56 29 


.07810 


.11970 


.09839 


.12543 


.11814 


.13126 


.13739 


.13721 


s ' 


2 h 42 40 


2 h 4& m 41 


2* 50 m 42 


2* 54 m 43 


30 


9.07845 


0.11980 


9.09872 


0.12552 


9.11847 


0.13136 


9.13771 


0.13731 


4 31 


.07879 


.11989 


.09905 


.12562 


.11879 


.13146 


.13803 


.13741 


8 32 


.07913 


.11999 


.09939 


.12572 


.11912 


.13156 


.13834 


.13751 


^2 33 


.07947 


.12008 


.09972 


.12581 


.11944 


.13166 


.13866 


.13761 


/0 34 


.07981 


.12018 


.10005 


.12591 


.11977 


.13175 


.13898 


.13771 


20 35 


9.08016 


0.12027 


9.10039 


0.12600 


9.12009 


0.13185 


9.13929 


0.13781 


24 36 


.08050 


.12036 


.10072 


.12610 


.12041 


.13195 


.13961 


.13791 


28 37 


.08084 


.12046 


.10105 


.12620 


.12074 


.13205 


.13992 


.13801 


32 38 


.08118 


.12055 


.10138 


.12629 


.12106 


.13215 


.14024 


.13811 


30 39 


.08152 


.12065 


.10172 


.12639 


.12139 


.13225 


.14056 


.13822 


40 40 


9.08186 


0.12074 


9.10205 


0.12649 


9.12171 


0.13235 


9.14087 


0.13832 


44 41 


.08220 


.12084 


.10238 


.12658 


.12203 


.13244 


.14119 


.13842 


48 42 


.08254 


.12093 


.10271 


.12668 


.12236 


.13254 


.14150 


.13852 


52 43 


.08288 


.12103 


.10304 


.12678 


.12268 


.13264 


.14182 


.13862 


56 44 


.08323 


.12112 


.10337 


.12687 


.12300 


.13274 


.44213 


.13872 


s ' 


2 h 43 m 40 


2 h J^m. 41 


2 h 51 m 42 


2 h 55 m 43 


'/ 45 


9.08357 


0.12122 


9.10371 


0.12697 


9.12332 


0.13284 


9.14245 


0.13882 


4 46 


.08391 


.12131 


.10404 


.12707 


.12365 


.13294 


.14276 


.13892 


S 47 


.08425 


.12141 


.10437 


.12717 


.12397 


.13304 


.14307 


.13902 


.72 48 


.08459 


.12150 


.10470 


.12726 


.12429 


.13314 


.14339 


.13912 


16 49 


.08492 


.12160 


.10503 


.12736 


.12461 


.13323 


.14370 


.13922 


20 50 


9.08526 


0.12169 


9.10536 


0.12746 


9.12494 


0.13333 


9.14402 


0.13932 


24 51 


.08560 


.12179 


.10569 


.12755 


.12526 


.13343 


.14433 


.13942 


28 52 


.08594 


.12188 


.10602 


.12765 


.12558 


.13353 


.14465 


.13952 


32 53 


.08628 


.12198 


.10635 


.12775 


.12590 


.13363 


.14496 


.13962 


36 54 


.08662 


.12207 


.10668 


.12784 


.12622 


.13373 


.14527 


.13972 


40 55 


9.08696 


0.12217 


9.10701 


0.12794 


9.12655 


0.13383 


9.14559 


0.13983 


44 56 


.08730 


.12226 


.10734 


.12804 


.12687 


.13393 


.14590 


.13993 


48 57 


.08764 


.12236 


.10767 


.12814 


.12719 


.13403 


.14621 


.14003 


52 58 


.08797 


.12245 


.10800 


.12823 


.12751 


.13412 


.14653 


.14013 


56 59 


.08831 


.12255 


.10833 


.12833 


.12783 


.13422 


.14684 


.14023 


60 60 


9.08865 


12265 


9.10866 


0.12843 


9.12815 


0.13432 


9.14715 


0.14033 



Table 10. Haversine Table 



261 



s ' 


2* 5fi m 44 


3* O m 45 


SA 4 m 46 


3 h 8 m. 470 




Hav. 


No. 


Hav. 


No. 


Hav. 


No. 


Hav. 


No. 





9.14715 


0.14033 


9.16568 


0.14645 


9.18376 


0.15267 


9.20140 


0.15900 


4 1 


.14746 


.14043 


.16598 


.14655 


.18405 


.15278 


.20169 


.15911 


8 2 


.14778 


.14053 


.16629 


.14665 


.18435 


.15288 


.20198 


.15921 


12 3 


.14809 


.14063 


.16659 


.14676 


.18465 


.15298 


.20227 


.15932 


16 4 


.14840 


.14073 


.16690 


.14686 


.18495 


.15309 


.20256 


.15943 


20 5 


9.14871 


0.14084 


9.16720 


0.14696 


9.18524 


0.15319 


9.20285 


0.15953 


24 6 


.14902 


.14094 


.16751 


.14706 


.18554 


.15330 


.20314 


.15964 


28 7 


.14934 


.14104 


.16781 


.14717 


.18584 


.15340 


.20343 


.15975 


32 8 


.14965 


.14114 


.16812 


.14727 


.18613 


.15351 


.20372 


.15985 


36 9 


.14996 


.14124 


.16842 


.14737 


.18643 


.15361 


.20401 


.15996 


40 10 


9.15027 


0.14134 


9.16872 


0.14748 


9.18673 


0.15372 


9.20430 


0.16007 


44 11 


.15058 


.14144 


.16903 


.14758 


.18702 


.15382 


.20459 


.16017 


48 12 


.15089 


.14154 


.16933 


.14768 


.18732 


.15393 


.20488 


.16028 


52 13 


.15120 


.14165 


.16963 


.14779 


.18762 


.15403 


.20517 


.16039 


56 14 


.15152 


.14175 


.16994 


.14789 


.18791 


.15414 


.20546 


.16049 


s ' 


2 h a7 m 44 


3 h jm 45 


3k s m 46 


S h 9 m 470 


15 


9.15183 


0.14185 


9.17024 


0.14799 


9.18821 


0.15424 


9.20574 


0.16060 


4 16 


.15214 


.14195 


.17054 


.14810 


.18850 


.15435 


.20603 


.16071 


5 17 


.15245 


.14205 


.17085 


.14820 


.18880 


.15445 


.20632 


.16081 


12 18 


.15276 


.14215 


.17115 


.14830 


.18909 


.15456 


.20661 


.16092 


J6 19 


.15307 


.14226 


.17145 


.14841 


.18939 


.15466 


.20690 


.16103 


20 20 


9.15338 


0.14236 


9.17175 


0.14851 


9.18968 


0.15477 


9.20719 


0.16113 


24 21 


.15369 


.14246 


.17206 


.14861 


.18998 


.15487 


.20748 


.16124 


25 22 


.15400 


.14256 


.17236 


.14872 


.19027 


.15498 


.20776 


.16135 


32 23 


.15431 


.14266 


.17266 


.14882 


.19057 


.15509 


.20805 


.16146 


36 24 


.15462 


.14276 


.17296 


.14892 


.19086 


.15519 


.20834 


.16156 


40 25 


9.15493 


0.14287 


9.17327 


0.14903 


9.19116 


0.15530 


9.2Q863 


0.16167 


44 26 


.15524 


.14297 


.17357 


.14913 


.19145 


.15540 


.20891 


.16178 


4S 27 


.15555 


.14307 


.17387 


.14923 


.19175 


.15551 


.20920 


.16188 


52 28 


.15585 


.14317 


.17417 


.14934 


.19204 


.15561 


.20949 


.16199 


56 29 


.15616 


.14327 


.17447 


.14944 


.19234 


.15572 


.20978 


.16210 


s ' 


2* 58 >n 44 


3 A 2"* 45 


3 h ffn 45 


3h jo 47 


30 


9.15647 


0.14337 


9.17477 


0.14955 


9.19263 


0.15582 


9.21006 


0.16220 


4 31 


.15678 


.14348 


.17507 


.14965 


.19292 


.15593 


.21035 


.16231 


8 32 


.15709 


.14358 


.17538 


.14975 


.19322 


.15603 


.21064 


.16242 


12 33 


.15740 


.14368 


.17568 


.14986 


.19351 


.15614 


.21092 


.16253 


/6 34 


.15771 


.14378 


.17598 


.14996 


.19381 


.15625 


.21121 


.16263 


20 35 


9.15802 


0.14388 


9.17628 


0.15006 


9.19410 


0.15635 


9.21150 


0.16274 


24 36 


.15832 


.14399 


.17658 


.15017 


.19439 


.15646 


.21178 


.16285 


28 37 


.15863 


.14409 


.17688 


.15027 


.19469 


.15656 


.21207 


.16296 


32 38 


.15894 


.14419 


.17718 


.15038 


.19498 


.15667 


.21236 


.16306 


36 39 


.15925 


.14429 


.17748 


.15048 


.19527 


.15677 


.21264 


.16317 


40 40 


9.15955 


0.14440 


9.17778 


0.15058 


9.19557 


0.15688 


9.21293 


0.16328 


44 41 


.15986 


.14450 


.17808 


.15069 


.19586 


.15699 


.21322 


.16339 


48 42 


.16017 


.14460 


.17838 


.15079 


.19615 


.15709 


.21350 


.16349 


52 43 


.16048 


.14470 


.17868 


.15090 


.19644 


.15720 


.21379 


.16360 


56 44 


.16078 


.14480 


.17898 


.15100 


.19674 


.15730 


.21407 


.16371 


s ' 


2 A 5 m 44 


3 h S" 1 45 


3* 7 46 


3 h ll m 47 


45 


9.16109 


0.14491 


9.17928 


0.15110 


9.19703 


0.15741 


9.21436 


0.16382 


4 46 


.16140 


.14501 


.17958 


.15121 


.19732 


.15751 


.21464 


.16392 


S 47 


.16170 


.14511 


.17988 


.15131 


.19761 


.15762 


.21493 


.16403 


/2 48 


.16201 


.14521 


.18018 


.15142 


.19790 


.15773 


.21521 


.16414 


16 49 


.16232 


.14532 


.18048 


.15152 


.19820 


.15783 


.21550 


.16425 


20 50 


9.16262 


0.14542 


9.18077 


0.15163 


9.19849 


0.15794 


9.21578 


0.16436 


24 51 


.16293 


.14552 


.18107 


.15173 


.19878 


.15804 


.21607 


.16446 


28 52 


.16324 


.14562 


.18137 


.15183 


.19907 


.15815 


.21635 


.16457 


32 53 


.16354 


.14573 


.18167 


.15194 


.19936 


.15826 


.21664 


.16468 


36 54 


.16385 


.14583 


.18197 


.15204 


.19965 


.15836 


.21692 


.16479 


40 55 


9.16415 


0.14593 


9.18227 


0.15215 


9.19995 


0.15847 


9.21721 


0.16489 


44 56 


.16446 


.14604 


.18256 


.15225 


.20024 


.15858 


.21749 


.16500 


48 57 


.16476 


.14614 


.18286 


.15236 


.20053 


.15868 


.21778 


.16511 


52 58 


.16507 


.14624 


.18316 


.15246 


.20082 


.15879 


.21806 


.16522 


56 59 


.16537 


.14634 


.18346 


.15257 


.20111 


.15889 


.21834 


.16533 


60 60 


9.16568 


0.14645 


9.18376 


0.15267 


9.20140 


0.15900 


9.21863 


0.16543 



262 



Table 10. Haversine Table 



s ' 


3h 12 m 48 


3* 16 49 


3>> 20 m 50 


3* 24 m 51 




Hav. 


No. 


Hav. 


No. 


Hav. 


No. 


Hav. 


No. 





9.21863 


0.16543 


9.23545 


0.17197 


9.25190 


0.17861 


9.26797 


0.18534 


4 l 


.21891 


.16554 


.23573 


.17208 


.25217 


.17872 


.26823 


.18545 


8 2 


.21919 


.16565 


.23601 


.17219 


.25244 


.17883 


.26850 


.18557 


12 3 


.21948 


.16576 


.23629 


.17230 


.25271 


.17894 


.26876 


.18568 


16 4 


.21976 


.16587 


.23656 


.17241 


.25298 


.17905 


.26903 


.18579 


20 5 


9.22004 


0.16598 


9.23684 


0.17252 


9.25325 


0.17916 


9.26929 


0.18591 


24 6 


.22033 


.16608 


.23712 


.17263 


.25352 


.17928 


.26956 


.18602 


28 7 


.22061 


.16619 


.23739 


.17274 


.25379 


.17939 


.26982 


.18613 


32 8 


.22089 


.16630 


.23767 


.17285 


.25406 


.17950 


.27008 


.18624 


36 9 


.22118 


.16641 


.23794 


.17296 


.25433 


.17961 


.27035 


.18636 


40 10 


9.22146 


0.16652 


9.23822 


0.17307 


9.25460 


0.17972 


9.27061 


0.18647 


44 11 


.22174 


.16663 


.23850 


.17318 


.25487 


.17983 


.27088 


.18658 


48 12 


.22202 


.16673 


.23877 


.17329 


.25514 


.17995 


.27114 


.18670 


52 13 


.22231 


.16684 


.23905 


.17340 


.25541 


.18006 


.27140 


.18681 


56 14 


.22259 


.16695 


.23932 


.17351 


.25568 


.18017 


.27167 


.18692 


s ' 


3* 13 48 


3 h 17 m 49 


3 h 21 m 50 


3^ 25 m 51 


15 


9.22287 


0.16706 


9.23960 


0.17362 


9.25595 


0.18028 


9.27193 


0.18704 


4 16 


.22315 


.16717 


.23988 


.17373 


.25622 


.18039 


.27219 


.18715 


S 17 


.22343 


.16728 


.24015 


.17384 


.25649 


.18050 


.27246 


.18727 


12 18 


.22372 


.16738 


.24043 


.17395 


.25676 


.18062 


.27272 


.18738 


J<? 19 


.22400 


.16749 


.24070 


.17406 


.25703 


.18073 


.27298 


.18749 


20 20 


9.22428 


0.16760 


9.24098 


0.17417 


9.25729 


0.18084 


9.27325 


0.18761 


24 21 


.22456 


.16771 


.24125 


.17428 


.25756 


.18095 


.27351 


.18772 


25 22 


.22484 


.16782 


.24153 


.17439 


.25783 


.18106 


.27377 


.18783 


32 23 


.22512 


.16793 


.24180 


.17450 


.25810 


.18118 


.27403 


.18795 


3 24 


.22540 


.16804 


.24208 


.17461 


.25837 


.18129 


.27430 


.18806 


40 25 


9.22569 


0.16815 


9.24235 


0.17472 


9.25864 


0.18140 


9.27456 


0.18817 


44 26 


.22597 


.16825 


.24263 


.17483 


.25891 


.18151 


.27482 


.18829 


45 27 


.22625 


.16836 


.24290 


.17494 


.25917 


.18162 


.27508 


.18840 


52 28 


.22653 


.16847 


.24317 


.17505 


.25944 


.18174 


.27535 


.18852 


5 29 


.22681 


.16858 


.24345 


.17517 


.25971 


.18185 


.27561 


.18863 


s ' 


3h l^m 48 


3h is 49 


3 h 2%m 50 


3h 26 51 


30 


9.22709 


0.16869 


9.24372 


0.17528 


9.25998 


0.18196 


9.27587 


0.18874 


4 31 


.22737 


.16880 


.24400 


.17539 


.26025 


.18207 


.27613 


.18886 


8 32 


.22765 


.16891 


.24427 


.17550 


.26051 


.18219 


.27639 


.18897 


^2 33 


.22793 


.16902 


.24454 


.17561 


.26078 


.18230 


.27666 


.18908 


Iff 34 


.22821 


.16913 


.24482 


.17572 


.26105 


.18241 


.27692 


.18920 


20 35 


9.22849 


0.16924 


9.24509 


0.17583 


9.26132 


0.18252 


9.27718 


0.18931 


24 36 


.22877 


.16934 


.24536 


.17594 


.26158 


.18263 


.27744 


.18943 


28 37 


.22905 


.16945 


.24564 


.17605 


.26185 


.18275 


.27770 


.18954 


32 38 


.22933 


.16956 


.24591 


.17616 


.26212 


.18286 


.27796 


.18965 


3 39 


.22961 


.16967 


.24618 


.17627 


.26238 


.18297 


.27822 


.18977 


40 40 


9.22989 


0.16978 


9.24646 


0.17638 


9.26265 


0.18308 


9.27848 


0.18988 


44 41 


.23017 


.16989 


.24673 


.17649 


.26292 


.18320 


.27875 


.19000 


48 42 


.23045 


.17000 


.24700 


.17661 


.26319 


.18331 


.27901 


.19011 


52 43 


.23073 


.17011 


.24728 


.17672 


.26345 


.18342 


.27927 


.19022 


5 44 


.23100 


.17022 


.24755 


.17683 


.26372 


.18353 


.27953 


.19034 


s ' 


3* 15 m 48 


gh 10m 49 


S h 23 50 


3 h 27 m 51 u 


45 


9.23128 


0.17033 


9.24782 


0.17694 


9.26398 


0.18365 


9.27979 


0.19045 


4 46 


.23156 


.17044 


.24809 


.17705 


.26425 


.18376 


.28005 


.19057 


S 47 


.23184 


.17055 


.24837 


.17716 


.26452 


.18387 


.28031 


.19068 


J2 48 


.23212 


.17066 


.24864 


.17727 


.26478 


.18399 


.28057 


.19080 


16 49 


.23240 


.17076 


.24891 


.17738 


.26505 


.18410 


.28083 


.19091 


20 50 


9.23268 


0.17087 


9.24918 


0.17749 


9.26532 


0.18421 


9.28109 


0.19102 


24 51 


.23295 


.17098 


.24945 


.17760 


.26558 


.18432 


.28135 


.19114 


28 52 


.23323 


.17109 


.24973 


.17772 


.26585 


.18444 


.28161 


.19125 


32 53 


.23351 


.17120 


.25000 


.17783 


.26611 


.18455 


.28187 


.19137 


36 54 


.23379 


.17131 


.25027 


.17794 


.26638 


.18466 


.28213 


.19148 


40 55 


9.23407 


0.17142 


9.25054 


0.17805 


9.26664 


0.18478 


9.28239 


0.19160 


44 56 


.23434 


.17153 


.25081 


.17816 


.26691 


.18489 


.28265 


.19171 


48 57 


.23462 


.17164 


.25108 


.17827 


.26717 


.18500 


.28291 


.19183 


52 58 


.23490 


.17175 


.25135 


.17838 


.26744 


.18511 


.28317 


.19194 


56 59 


.23518 


.17186 


.25163 


.17849 


.26770 


.18523 


.28342 


.19205 


50 60 


9.23545 


0.17197 


9.25190 


0.17861 


9.26797 


0.18534 


9.28368 


0.19217 



Table 10. Haversine Table 



263 



s ' 


Sh 28 52 


3* 32 m 53 


Sh 36 54 


3* 40 m 55 




Bvr. 


No. 


Hav. 


No. 


Hav. 


No. 


Hav. 


No. 





).2S3(iS 


0.19217 


9.29906 


0.19909 


9.31409 


0.20611 


9.32881 


021321 


4 1 


.28394 


.19228 


.29931 


.19921 


.31434 


.20623 


.32905 


.21333 


8 2 


.28420 


.19240 


.29956 


.19932 


.31459 


.20634 


.32930 


.21345 


12 3 


.28446 


.19251 


.29981 


.19944 


.31484 


.20646 


.32954 


.21357 


16 4 


.28472 


.19263 


.30007 


.19956 


.31508 


.20658 


.32978 


.21369 


20 5 


9.28498 


0.19274 


9.30032 


0.19967 


9.31533 


0.20670 


9.33002 


0.21381 


24 6 


.28524 


.19286 


.30057 


.19979 


.31558 


.20681 


.33027 


.21393 


28 7 


.28549 


.19297 


.30083 


.19991 


.31583 


.20693 


.33051 


.21405 


32 8 


.28575 


.19309 


.30108 


.20002 


.31607 


.20705 


.33075 


.21417 


86 9 


.28601 


.19320 


.30133 


.20014 


.31632 


.20717 


.33099 


.21429 


40 10 


9.28627 


0.19332 


9.30158 


0.20026 


9.31657 


0.20729 


9.33123 


0.21440 


-a 11 


.28653 


.19343 


.30184 


.20037 


.31682 


.20740 


.33148 


.21452 


48 12 


.28679 


.19355 


.30209 


.20049 


.31706 


.20752 


.33172 


.21464 


52 13 


.28704 


.19366 


.30234 


.20060 


.31731 


.20764 


.33196 


.21476 


56 14 


.28730 


.19378 


.30259 


.20072 


.31756 


.20776 


.33220 


.21488 


s ' 


3 h 2ST 52 


3 h 33 53 


Sh 37^ 54 


3 h 41 m 55 


15 


9.28756 


0.19389 


9.30285 


0.20084 


9.31780 


0.20788 


9.33244 


0.21500 


4 16 


.28782 


.19401 


.30310 


.20095 


.31805 


.20799 


.33268 


.21512 


S 17 


.28807 


.19412 


.30335 


.20107 


.31830 


.20811 


.33292 


.21524 


12 18 


.28833 


.19424 


.30360 


.20119 


.31854 


.20823 


.33317 


.21536 


J<? 19 


.28859 


.19435 


.30385 


.20130 


.31879 


.20835 


.33341 


.21548 


20 20 


9.28885 


0.19447 


9.30410 


0.20142 


9.31903 


0.20847 


9.33365 


0.21560 


24 21 


.28910 


.19458 


.30436 


.20154 


.31928 


.20858 


.33389 


.21572 


25 22 


.28936 


.19470 


.30461 


.20165 


.31953 


.20870 


.33413 


.21584 


32 23 


.28962 


.19481 


.30486 


.20177 


.31977 


.20882 


.33437 


.21596 


35 24 


.28987 


.19493 


.30511 


.20189 


.32002 


.20894 


.33461 


.21608 


40 25 


9.29013 


0.19504 


9.30536 


0.20200 


9.32026 


0.20906 


9.33485 


0.21620 


44 26 


.29039 


.19516 


.30561 


.20212 


.32051 


.20918 


.33509 


.21632 


45 27 


.29064 


.19527 


.30586 


.20224 


.32076 


.20929 


.33533 


.21644 


52 28 


.29090 


.19539 


.30611 


.20235 


.32100 


.20941 


.33557 


.21656 


55 29 


.29116 


.19550 


.30636 


.20247 


.32125 


.20953 


.33581 


.21668 


s ' 


3* 30 m 52 


gh 3jm 53 


Sh 38^ 54 


3 h 42"' 55 


30 


9.29141 


0.19562 


9.30662 


0.20259 


9.32149 


0.20965 


9.33605 


0.21680 


4 31 


.29167 


.19573 


.30687 


.20271 


.32174 


.20977 


.33629 


.21692 


8 32 


.29192 


.19585 


.30712 


.20282 


.32198 


.20989 


.33653 


.21704 


72 33 


.29218 


.19597 


.30737 


.20294 


.32223 


.21000 


.33677 


.21716 


76 34 


.29244 


.19608 


.30762 


.20306 


.32247 


.21012 


.33701 


.21728 


20 35 


9.29269 


0.19620 


9.30787 


0.20317 


9.32272 


0.21024 


9.33725 


0.21740 


24 36 


.29295 


.19631 


.30812 


.20329 


.32296 


.21036 


.33749 


.21752 


28 37 


.29320 


.19643 


.30837 


.20341 


.32321 


.21048 


.33773 


.21764 


32 38 


.29346 


.19654 


.30862 


.20352 


.32345 


.21060 


.33797 


.21776 


36 39 


.29371 


.19666 


.30887 


.20364 


.32370 


.21072 


.33821 


.21788 


40 40 


9.29397 


0.19677 


9.30912 


0.20376 


9.32394 


0.21083 


9.33845 


0.21800 


44 41 


.29422 


.19689 


.30937 


.20388 


.32418 


.21095 


.33869 


.21812 


48 42 


.29448 


.19701 


.30962 


.20399 


.32443 


.21107 


.33893 


.21824 


52 43 


.29473 


.19712 


.30987 


.20411 


.32467 


.21119 


.33917 


.21836 


55 44 


.29499 


.19724 


.31012 


.20423 


.32492 


.21131 


.33941 


.21848 


s ' 


3h sim 52 


3* 35 m 53 


3 h 39 54 


Sh 43" 55 


45 


9.29524 


0.19735 


9.31036 


0.20435 


9.32516 


0.21143 


9.33965 


0.21860 


4 46 


.29550 


.19747 


.31061 


.20446 


.32541 


.21155 


.33988 


.21872 


S 47 


.29575 


.19758 


.31086 


.20458 


.32565 


.21167 


.34012 


.21884 


12 48 


.29601 


.19770 


.31111 


.20470 


.32589 


.21178 


.34036 


.21896 


16 49 


.29626 


.19782 


.31136 


.20481 


.32614 


.21190 


.34060 


.21908 


20 50 


9.29652 


0.19793 


9.31161 


0.20493 


9.32638 


0.21202 


9.34084 


0.21920 


24 51 


.29677 


.19805 


.31186 


.20505 


.32662 


.21214 


.34108 


.21932 


28 52 


.29703 


.19816 


.31211 


.20517 


.32687 


.21226 


.34132 


.21944 


32 53 


.29728 


.19828 


.31236 


.20528 


.32711 


.21238 


.34155 


.21956 


36 54 


.29753 


.19840 


.31260 


.20540 


.32735 


.21250 


.34179 


.21968 


40 55 


9.29779 


0.19851 


9.31285 


0.20552 


9.32760 


0.21262 


9.34203 


0.21980 


44 56 


.29804 


.19863 


.31310 


.20564 


.32784 


.21274 


.34227 


.21992 


48 57 


.29829 


.19874 


.31335 


.20575 


.32808 


.21285 


.34251 


.22004 


52 58 


.29855 


.19886 


.31360 


.20587 


.32833 


.21297 


.34274 


.22016 


56 59 


.29880 


.19898 


.31385 


.20599 


.32857 


.21309 


.34298 


.22028 


(SO 60 


9.29906 


0.19909 


9.31409 


0.20611 


9.32881 


0.21321 


9.34322 


0.22040 



264 



Table 10. Haversine Table 



, 


SA 44 m 56 


gA 48 57 


3* 52 58 


3* 56 59 




Hav. 


No. 


Hav. 


No. 


Hav. 


No. 


Hav. 


No. 





9.34322 


0.22040 


9.35733 


0.22768 


9.37114 


0.23504 


9.38468 


0.24248 


4 1 


.34346 


.22052 


.35756 


.22780 


.37137 


.23516 


.38490 


.24261 


8 2 


.34369 


.22064 


.35779 


.22792 


.37160 


.23529 


.38512 


.24273 


12 3 


.34393 


.22077 


.35802 


.22805 


.37183 


.23541 


.38535 


.24286 


16 4 


.34417 


.22089 


.35826 


.22817 


.37205 


.23553 


.38557 


.24298 


20 5 


9.34441 


0.22101 


9.35849 


0.22829 


9.37228 


0.23566 


9.38579 


0.24310 


24 6 


.34464 


.22113 


.35872 


.22841 


.37251 


.23578 


.38602 


.24323 


28 1 


.34488 


.22125 


.35895 


.22853 


.37274 


.23590 


.38624 


.24335 


32 8 


.34512 


.22137 


.35918 


.22866 


.37296 


.23603 


.38646 


.24348 


Sff 9 


.34535 


.22149 


.35942 


.22878 


.37319 


.23615 


.38668 


.24360 


40 10 


9.34559 


0.22161 


9.35965 


0.22890 


9.37342 


0.23627 


9.38691 


0.24373 


44 11 


.34583 


.22173 


.35988 


.22902 


.37364 


.23640 


.38713 


.24385 


48 12 


.34606 


.22185 


.36011 


.22915 


.37387 


.23652 


.38735 


.24398 


52 13 


.34630 


.22197 


.36034 


.22927 


.37410 


.23665 


.38757 


.24410 


off 14 


.34654 


.22209 


.36058 


.22939 


.37433 


.23677 


.38780 


.24423 


s ' 


3* 45 m 56 


3h 49 m 57 


3* 5S" 1 58 


3h 57 59 


15 


9.34677 


0.22221 


9.36081 


0.22951 


9.37455 


0.23689 


9.38802 


0.24435 


4 16 


.34701 


.22234 


.36104 


.22964 


.37478 


.23702 


.38824 


.24448 


5 17 


.34725 


.22246 


.36127 


.22976 


.37501 


.23714 


.38846 


.24460 


12 18 


.34748 


.22258 


.36150 


.22988 


.37523 


.23726 


.38868 


.24473 


16 19 


.34772 


.22270 


.36173 


.23000 


.37546 


.23739 


.38891 


.24485 


20 20 


9.34795 


0.22282 


9.36196 


0.23012 


9.37569 


0.23751 


9.38913 


0.24498 


24. 21 


.34819 


.22294 


.36219 


.23025 


.37591 


.23764 


.38935 


.24510 


25 22 


.34843 


.22306 


.36243 


.23037 


.37614 


.23776 


.38957 


.24523 


32 23 


.34866 


.22318 


.36266 


.23049 


.37636 


.23788 


.38979 


.24535 


36 24 


.34890 


.22330 


.36289 


.23061 


.37659 


.23801 


.39002 


.24548 


40 25 


9.34913 


0.22343 


9.36312 


0.23074 


9.37682 


0.23813 


9.39024 


0.24560 


44 26 


.34937 


.22355 


.36335 


.23086 


.37704 


.23825 


.39046 


.24573 


48 27 


.34960 


.22367 


.36358 


.23098 


.37727 


.23838 


.39068 


.24586 


52 25 


.34984 


.22379 


.36381 


.23110 


.37749 


.23850 


.39090 


.24598 


56 29 


.35007 


.22391 


.36404 


.23123 


.37772 


.23863 


.39112 


.24611 


s ' 


3* 46 56 


3h 50 m 57 


3* 54 m 58 


3 h 58 59 


30 


9.35031 


0.22403 


9.36427 


0.23135 


9.37794 


0.23875 


9.39134 


0.24623 


4 31 


.35054 


.22415 


.36450 


.23147 


.37817 


.23887 


.39156 


.24636 


5 32 


.35078 


.22427 


.30473 


.23160 


.37840 


.23900 


.39178 


.24648 


12 33 


.35101 


.22440 


.36496 


.23172 


.37862 


.23912 


.39201 


24661 


J6 34 


.35125 


.22452 


.36519 


.23184 


.37885 


.23925 


.39223 


.24673 


20 35 


9.35148 


0.22464 


9.36542 


0.23196 


9.37907 


0.23937 


9.39245 


0.24686 


24 36 


.35172 


.22476 


.36565 


.23209 


.37930 


.23950 


.39267 


.24698 


25 37 


.35195 


.22488 


.36588 


.23221 


.37952 


.23962 


.39289 


.24711 


32 38 


.35219 


.22500 


.36611 


.23233 


.37975 


.23974 


.39311 


.24723 


36 39 


.35242 


.22512 


.36634 


.23246 


.37997 


.23987 


.39333 


.24736 


40 40 


9.35266 


0.22525 


9.36657 


0.23258 


9.38020 


0.23999 


9.39355 


0.24749 


44 41 


.35289 


.22537 


.36680 


.23270 


.38042 


.24012 


.39377 


.24761 


45 42 


.35312 


.22549 


.36703 


.23282 


.38065 


.24024 


.39399 


.24774 


52 43 


.35336 


.22561 


.36726 


.23295 


.38087 


.24036 


.39421 


.24786 


56 44 


.35359 


.22573 


.36749 


.23307 


.38110 


.24049 


39443 


.24799 


s ' 


3* 47 m 56 


3 h 51 m 57 


3h 55m 5 8 


3h SQ 59 


45 


9.35383 


0.22585 


9.36772 


0.23319 


9.38132 


0.24061 


9.39465 


0.24811 


4 46 


.35406 


.22598 


.36794 


.23332 


.38154 


.24074 


.39487 


.24824 


5 47 


.35429 


.22610 


.36817 


.23344 


.38177 


.24086 


.39509 


.24836 


12 48 


.35453 


.22622 


.36840 


.23356 


.38199 


.24099 


.39531 


.24849 


16 49 


.35476 


.22634 


.36863 


.23368 


.38222 


.24111 


.39553 


.24862 


20 50 


9.35500 


0.22646 


9.36886 


0.23381 


9.38244 


0.24124 


9.39575 


0.24874 


24 51 


.35523 


.22658 


.36909 


.23393 


.38267 


.24136 


.39597 


.24887 


25 52 


.35546 


.22671 


.36932 


.23405 


.38289 


.24148 


.39619 


.24899 


32 53 


.35570 


.22683 


.36955 


.23418 


.38311 


.24161 


.39641 


.24912 


36 54 


.35593 


.22695 


.36977 


.23430 


.38334 


.24173 


.39663 


.24924 


40 55 


9.35616 


0.22707 


9.37000 


0.23442 


9.38356 


0.24186 


9.39685 


0.24937 


44 56 


.35639 


.22719 


.37023 


.23455 


.38378 


.24198 


.39706 


.24950 


45 57 


.35663 


.22731 


.37046 


.23467 


.38401 


.24211 


.39728 


.24962 


52 58 


.35686 


.22744 


.37069 


.23479 


.38423 


.24223 


.39750 


.24975 


56 59 


.35709 


.22756 


.37091 


.23492 


.38445 


.24236 


.39772 


.24987 


50 60 


9.35733 


0.22768 


9.37114 


0.235C4 


9.38468 


0.24248 


9.39794 


0.25000 



Table 10. Hayersine Table 



265 



s ' 


4* O m 60 


4 h 4 m 61 


4* 8 m 62 


4* 12 63 




Hav. 


No. 


Uav. 


No. 


Hav. 


No. 


Hav. 


No. 





9.39794 


0.25000 


9.41094 


0.25760 


9.42368 


0.26526 


9.43617 


0.27300 


4 1 


.39816 


.25013 


.41115 


.25772 


.42389 


.26539 


.43638 


.27313 


8 2 


.39838 


.25025 


.41137 


.25785 


.42410 


.26552 


.43658 


.27326 


12 3 


.39860 


.25038 


.41158 


.25798 


.42431 


.26565 


.43679 


.27339 


16 4 


.39881 


.25050 


.41180 


.25810 


.42452 


.26578 


.43699 


.27352 


20 5 


9.39903 


0.25063 


9.41201 


0.25823 


9.42473 


0.26591 


9.43720 


0.27365 


24 6 


.39925 


.25076 


.41222 


.25836 


.42494 


.26604 


.43741 


.27378 


28 7 


.39947 


.25088 


.41244 


.25849 


.42515 


.26616 


.43761 


.27391 


32 8 


.39969 


.25101 


.41265 


.25861 


.42536 


.26629 


.43782 


.27404 


36 9 


.39991 


.25113 


.41287 


.25874 


.42557 


.26642 


.43802 


.27417 


40 10 


9.40012 


0.25126 


9.41308 


0.25887 


9.42578 


0.26655 


9.43823 


0.27430 


44 11 


.40034 


.25139 


.41329 


.25900 


.42599 


.26668 


.43843 


.27443 


48 12 


.40056 


.25151 


.41351 


.25912 


.42620 


.26681 


.43864 


.27456 


52 13 


.40078 


.25164 


.41372 


.25925 


.42641 


.26694 


.43884 


.27469 


56 14 


.40100 


.25177 


.41393 


.25938 


.42662 


.26706 


.43905 


.27482 


s ' 


4* l m 60 


4h 5 m 61 


4* 9 m 62 


4* 13 m 63 


15 


9.40121 


0.25189 


9.41415 


0.25951 


9.42682 


0.26719 


9.43926 


0.27495 


4 16 


.40143 


.25202 


.41436 


.25963 


.42703 


.26732 


.43946 


.27508 


S 17 


.40165 


.25214 


.41457 


.25976 


.42724 


.26745 


.43967 


.27521 


12 18 


.40187 


.25227 


.41479 


.25989 


.42745 


.26758 


.43987 


.27534 


/<? 19 


.40208 


.25240 


.41500 


.26002 


.42766 


.26771 


.44008 


.27547 


20 20 


9.40230 


0.25252 


9.41521 


0.26014 


9.42787 


0.26784 


9.44028 


0.27560 


24 21 


.40252 


.25265 


.41543 


.26027 


.42808 


.26797 


.44048 


.27573 


2<S 22 


.40274 


.25278 


.41564 


.26040 


.42829 


.26809 


.44069 


.27586 


32 23 


.40295 


.25290 


.41585 


.26053 


.42850 


.26822 


.44089 


.27599 


36 24 


.40317 


.25303 


.41606 


.26065 


.42870 


.26835 


.44110 


.27612 


40 25 


9.40339 


0.25316 


9.41628 


0.26078 


9.42891 


0.26848 


9.44130 


0.27625 


44 26 


.40360 


.25328 


.41649 


.26091 


.42912 


.26861 


.44151 


.27638 


45 27 


.40382 


.25341 


.41670 


.26104 


.42933 


.26874 


.44171 


.27651 


52 28 


.40404 


.25354 


.41692 


.26117 


.42954 


.26887 


.44192 


.27664 


56 29 


.40425 


.25366 


.41713 


.26129 


.42975 


.26900 


.44212 


.27677 


s ' 


4* 2 60 


4 6 61 


4k 10 62 


4* I4 m 63 


30 


9.40447 


0.25379 


9.41734 


0.26142 


9.42996 


0.26913 


9.44232 


0.27690 


4 31 


.40469 


.25391 


.41755 


.26155 


.43016 


.26925 


.44253 


.27703 


8 32 


.40490 


.25404 


.41776 


.26168 


.43037 


.26938 


.44273 


.27716 


J2 33 


.40512 


.25417 


.41798 


.26180 


.43058 


.26951 


.44294 


.27729 


/'/ 34 


.40534 


.25429 


.41819 


.26193 


.43079 


.26964 


.44314 


.27742 


20 35 


9.40555 


0.25442 


9.41840 


0.26206 


9.43100 


0.26977 


9.44334 


0.27755 


24 36 


.40577 


.25455 


.41861 


.26219 


.43120 


.26990 


.44355 


.27768 


28 37 


.40599 


.25467 


.41882 


.26232 


.43141 


.27003 


.44375 


.27781 


32 38 


.40620 


.25480 


.41904 


.26244 


.43162 


.27016 


.44396 


.27794 


36 39 


.40642 


.25493 


.41925 


.26257 


.43183 


.27029 


.44416 


.27807 


40 40 


9.40663 


0.25506 


9.41946 


0.26270 


9.43203 


0.27042 


9.44436 


0.27820 


44 41 


.40685 


.25518 


.41967 


.26283 


.43224 


.27055 


.44457 


.27833 


48 42 


.40707 


.25531 


.41988 


.26296 


.43245 


.27068 


.44477 


.27846 


52 43 


.40728 


.25544 


.42009 


26308 


.43266 


.27080 


.44497 


.27859 


56 44 


.40750 


.25556 


.42031 


.26321 


.43286 


.27093 


.44518 


.27873 


8 ' 


4*3- 60 


4* 7 m 61 


4* ll m 62 


4* I5 m 63 


45 


9.40771 


0.25569 


9.42052 


0.26334 


9.43307 


0.27106 


9.44538 


0.27886 


4 46 


.40793 


.25582 


.42073 


.26347 


.43328 


.27119 


.44558 


.27899 


S 47 


.40814 


.25594 


.42094 


26360 


.43348 


.27132 


.44579 


.27912 


/2 48 


.40836 


.25607 


.42115 


.26372 


.43369 


.27145 


.44599 


.27925 


16 49 


.40858 


.25620 


.42136 


.26385 


.43390 


.27158 


.44619 


.27938 


20 50 


9.40879 


025632 


9.42157 


0.26398 


9.43411 


0.27171 


9.44639 


0.27951 


24 51 


.40900 


.25645 


.42178 


.26411 


.43431 


.27184 


.44660 


.27964 


2S 52 


.40922 


.25658 


.42199 


.26424 


.43452 


.27197 


.44680 


.27977 


32 63 


.40943 


.25671 


.42221 


.26437 


.43473 


.27210 


.44700 


.27990 


36 54 


.40965 


.25683 


.42242 


.26449 


.43493 


.27223 


.44721 


.28003 


40 55 


9.40986 


0.25696 


9.42263 


0.26462 


9.43514 


0.27236 


9.44741 


0.28016 


44 66 


.41008 


.25709 


.42284 


.26475 


.43535 


.27249 


.44761 


.28029 


48 57 


.41029 


.25721 


.42305 


.26488 


.43555 


.27262 


.44781 


.28042 


52 58 


.41051 


.25734 


.42326 


.26501 


.43576 


.27275 


.44801 


.28055 


->H 59 


.41072 


.25747 


.42347 


.26514 


.43596 


.27288 


.44822 


.28068 


rt f,c. 


Q 4 1 HO/! 


n 9R7Afl 


Q /lOQftC 


n QCCOC 


O AtRf7 


n OTinn 


1 1 1 v !> 


n 9ni 



266 



Table 10. Haversine Table 



s ' 


4* K?" 1 64 


4* 20 m 65 


4* 24 m . 66 


4 h 28 67 




Hav. 


No. 


Hav. 


No. 


Hav. 


No. 


Hav. 


No. 





9.44842 


0.28081 


9.46043 


0.28869 


9.47222 


0.29663 


9.48378 


0.30463 


4 1 


.44862 


.28095 


.46063 


.28882 


.47241 


.29676 


.48397 


.30477 


8 2 


.44882 


.28108 


.46083 


.28895 


.47261 


.29690 


.48416 


.30490 


12 3 


.44903 


.28121 


.46103 


.28909 


.47280 


.29703 


.48435 


.30504 


16 4 


.44923 


.28134 


.46123 


.28922 


.47300 


.29716 


.48454 


.30517 


20 5 


9.44943 


0.28147 


9.46142 


0.28935 


9.47319 


0.29730 


9.48473 


0.30530 


24 6 


.44963 


.28160 


.46162 


.28948 


.47338 


.29743 


.48492 


.30544 


28 7 


.44983 


.28173 


.46182 


.28961 


.47358 


.29756 


.48511 


.30557 


32 8 


.45003 


.28186 


.46202 


.28975 


.47377 


.29770 


.48530 


.30571 


36 9 


.45024 


.28199 


.46222 


.28988 


.47397 


.29783 


.48549 


.30584 


40 10 


9.45044 


0.28212 


9.46241 


0.29001 


9.47416 


0.29796 


9.48568 


0.30597 


44 11 


.45064 


.28225 


.46261 


.29014 


.47435 


.29809 


.48587 


.30611 


48 12 


.45084 


.28238 


.46281 


.29027 


.47455 


.29823 


.48607 


.30624 


52 13 


.45104 


.28252 


.46301 


.29041 


.47474 


.29836 


.48626 


.30638 


56 14 


.45124 


.28265 


.46320 


.29054 


.47493 


.29849 


.48645 


.30651 


s ' 


4* 17 m 64 


4 h 21 m 65 


4* 25'" 66 


4* 29 67 


15 


9.45144 


0.28278 


9.46340 


0.29067 


9.47513 


0.29863 


9.48664 


0.30664 


4 16 


.45165 


.28291 


.46360 


.29080 


.47532 


.29876 


.48683 


.30678 


S 17 


.45185 


.28304 


.46380 


.29093 


.47552 


.29889 


.48702 


.30691 


12 18 


.45205 


.28317 


.46399 


.29107 


.47571 


.29903 


.48720 


.30705 


1 19 


.45225 


.28330 


.46419 


.29120 


.47590 


.29916 


.48739 


.30718 


20 20 


9.45245 


0.28343 


9.46439 


0.29133 


9.47610 


0.29929 


9.48758 


0.30732 


24 21 


.45265 


.28356 


.46458 


.29146 


.47629 


.29943 


.48777 


.30745 


25 22 


.45285 


.28369 


.46478 


.29160 


.47648 


.29956 


.48796 


.30758 


32 23 


.45305 


.28383 


.46498 


.29173 


.47668 


.29969 


.48815 


.30772 


3 24 


.45325 


.28396 


.46517 


.29186 


.47687 


.29983 


.48834 


.30785 


40 25 


9.45345 


0.28409 


9.46537 


0.29199 


9.47706 


0.29996 


9.48853 


0.30799 


44 26 


.45365 


.28422 


.46557 


.29212 


.47725 


.30009 


.48872 


.30812 


45 27 


.45385 


.28435 


.46576 


.29226 


.47745 


.30023 


.48891 


.30826 


52 28 


.45405 


.28448 


.46596 


.29239 


.47764 


.30036 


.48910 


.30839 


56 29 


.45426 


.28461 


.46616 


.29252 


.47783 


.30049 


.48929 


.30852 


s ' 


4* 18 64 


4 h 22 65 


4* 26 66 


4* SO 67 


30 


9.45446 


0.28474 


9.46635 


0.29265 


9.47803 


0.30063 


9.48948 


0.30866 


4 31 


.45466 


.28488 


.46655 


.29279 


.47822 


.30076 


.48967 


.30879 


8 32 


.45486 


.28501 


.46675 


.29292 


.47841 


.30089 


.48986 


.30893 


12 33 


.45506 


.28514 


.46694 


.29305 


.47860 


.30103 


.49004 


.30906 


1<S 34 


.45526 


.28527 


.46714 


.29318 


.47880 


.30116 


.49023 


.30920 


20 35 


9.45546 


0.28540 


9.46733 


0.29332 


9.47899 


0.30129 


9.49042 


0.30933 


24 36 


.45566 


.28553 


.46753 


.29345 


.47918 


.30143 


.49061 


.30946 


28 37 


.45586 


.28566 


.46773 


.29358 


.47937 


.30156 


49080 


.30960 


32 38 


.45606 


.28580 


.46792 


.29371 


.47957 


.30169 


.49099 


.30973 


36 39 


.45625 


.28593 


.46812 


.29385 


.47976 


.30183 


.49118 


.30987 


40 40 


9.45645 


0.28606 


9.46831 


0.29398 


9.47995 


0.30196 


9.49137 


0.31000 


44 41 


.45665 


.28619 


.46851 


.29411 


.48014 


.30209 


.49155 


.31014 


48 42 


.45685 


.28632 


.46871 


.29424 


.48033 


.30223 


.49174 


.31027 


52 43 


.45705 


.28645 


.46890 


.29438 


.48053 


.30236 


.49193 


.31041 


56 44 


.45725 


.28658 


.46910 


.29451 


.48072 


.30249 


9212 


.31054 


s ' 


4* 19 64 


4* 23 65 


4* 27 66 


4* 31 67 


45 


9.45745 


0.28672 


9.46929 


0.29464 


9.48091 


0.30263 


9.49231 


0.31068 


4 46 


.45765 


.28685 


.46949 


.29477 


.48110 


.30276 


.49250 


.31081 


5 47 


.45785 


.28698 


.46968 


.29491 


.48129 


.30290 


.49268 


.31095 


12 48 


.45805 


.28711 


.46988 


.29504 


.48148 


.30303 


.49287 


.31108 


16 49 


.45825 


.28724 


.47007 


.29517 


.48168 


.30316 


.49306 


.31121 


20 50 


9.45845 


0.28737 


9.47027 


0.29530 


9.48187 


0.30330 


9.49325 


0.31135 


24 51 


.45865 


.28751 


.47046 


.29544 


.48206 


.30343 


.49344 


.31148 


28 52 


.45884 


.28764 


.47066 


.29557 


.48225 


.30356 


.49362 


.31162 


32 53 


.45904 


.28777 


.47085 


.29570 


.48244 


.30370 


.49481 


.31175 


36 54 


.45924 


.28790 


.47105 


.29583 


.48263 


.30383 


.49400 


.31189 


40 55 


9.45944 


0.28803 


9.47124 


0.29597 


9.48282 


0.30397 


9.49419 


0.31202 


44 56 


.45964 


.28816 


.47144 


.29610 


.48302 


.30410 


.49437 


.31216 


48 57 


.45984 


.28830 


.47163 


.29623 


.48321 


.30423 


.49456 


.31229 


52 58 


.46004 


.28843 


,47183 


.29637 


.48340 


.30437 


.49475 


.31243 


56 59 


.46023 


.28856 


.47202 


.29650 


.48359 


.30450 


.49494 


.31256 


60 60 


9.46043 


0.28869 


9.47222 


0.29663 


9.48378 


0.30463 


9.49512 


0.31270 



Table 10. Haversine Table 



26' 



s ' 


4* S2 m 68 


4* 36 69 


4* 4Q m 70 


4* 44 m 71 




Hav. 


No. 


Hav. 


No. 


Hav. 


No. 


Hav. 


No. 





9.49512 


0.31270 


9.50626 


0.32082 


9.51718 


0.32899 


9.52791 


0.33722 


4 1 


.49531 


.31283 


.50644 


.32095 


.51736 


.32913 


.52809 


.33735 


8 2 


.49550 


.31297 


.50662 


.32109 


.51754 


.32926 


.52826 


.33749 


12 3 


.49568 


.31310 


.50681 


.32122 


.51772 


.32940 


.52844 


.33763 


16 4 


.49587 


.31324 


.50699 


.32136 


.51790 


.32954 


.52862 


.33777 


20 5 


9.49606 


0.31337 


9.50717 


0.32150 


9.51808 


0.32967 


9.52879 


0.33790 


24 6 


.49625 


.31351 


.50736 


.32163 


.51826 


.32981 


.52897 


.33804 


28 7 


.49643 


.31364 


.50754 


.32177 


.51844 


.32995 


.52915 


.33818 


32 8 


.49662 


.31378 


.50772 


.32190 


.51862 


.33008 


.52932 


.33832 


36 9 


.49681 


.31391 


.50791 


.32204 


.51880 


.33022 


.52950 


.33845 


40. 10 


9.49699 


0.31405 


9.50809 


0.32217 


9.51898 


0.33036 


9.52968 


0.33859 


44 11 


.49718 


.31418 


.50827 


.32231 


.51916 


.33049 


.52985 


.33873 


48 12 


.49737 


.31432 


.50846 


.32245 


.51934 


.33063 


.53003 


.33887 


52 13 


.49755 


.31445 


.50864 


.32258 


.51952 


.33077 


.53021 


.33900 


56 14 


.49774 


.31459 


.50882 


.32272 


.51970 


.33090 


.53038 


.33914 


s ' 


4* 33"' 68 


4* 37 m 69 


4* 41 m 70 


4* 45 m 71 


15 


9.49793 


0.31472 


9.50901 


0.32285 


9.51988 


0.33104 


9.53056 


0.33928 


4 16 


.49811 


.31486 


.50919 


.32299 


.52006 


.33118 


.53073 


.33942 


S 17 


.49830 


.31499 


.50937 


.32313 


.52024 


.33132 


.53091 


.33956 


12 18 


.49849 


.31513 


.50956 


.32326 


.52042 


.33145 


.53109 


.33969 


76 19 


.49867 


.31526 


.50974 


.32340 


.52060 


.33159 


.53126 


.33983 


20 20 


9.49886 


0.31540 


9.50992 


0.32353 


9.52078 


0.33173 


9.53144 


0.33997 


24 21 


.49904 


.31553 


.51010 


.32367 


.52096 


.33186 


.53162 


.34011 


25 22 


.49923 


.31567 


.51029 


.32381 


.52114 


.33200 


.53179 


.34024 


32 23 


.49942 


.31580 


.51047 


.32394 


.52132 


.33214 


.53197 


.34038 


36 24 


.49960 


.31594 


.51065 


.32408 


.52150 


.33227 


.53214 


.34052 


40 25 


9.49979 


0.31607 


9.51083 


0.32422 


9.52168 


0.33241 


9.53232 


0.34066 


44 26 


.49997 


.31621 


.51102 


.32435 


.52185 


.33255 


.53249 


.34080 


4S 27 


.50016 


.31634 


.51120 


.32449 


.52203 


.33269 


.53267 


.34093 


.52 28 


.50034 


.31648 


.51138 


.32462 


.52221 


.33282 


.53285 


.34107 


5(5 29 


.50053 


.31661 


.51156 


.32476 


.52239 


.33296 


.53302 


.34121 


s ' 


4* S4 m 68 


4* 38 69 


4* 42 m 70 


4* 46 m 71 


30 


9.50072 


0.31675 


9.51174 


0.32490 


9.52257 


0.33310 


9.53320 


0.34135 


^ 31 


.50090 


.31688 


.51193 


.32503 


.52275 


.33323 


.53337 


.34149 


8 32 


.50109 


.31702 


.51211 


.32517 


.52293 


.33337 


.53355 


.34162 


/2 33 


.50127 


.31716 


.51229 


.32531 


.52311 


.33351 


.53372 


.34176 


76 34 


.50146 


.31729 


.51247 


.32544 


.52328 


.33365 


.53390 


.34190 


20 35 


9.50164 


0.31742 


9.51265 


0.32558 


9.52346 


0.33378 


9.53407 


0.34204 


24 36 


.50183 


.31756 


.51284 


.32571 


.52364 


.33392 


.53425 


.34218 


28 37 


.50201 


.31770 


.51302 


.32585 


.52382 


.33406 


.53442 


.34231 


32 38 


.50220 


.31783 


.51320 


.32599 


.52400 


.33419 


.53460 


.34245 


36 39 


.50238 


.31797 


.51338 


.32612 


.52418 


.33433 


.53477 


.34259 


40 40 


9.50257 


0.31810 


9.51356 


032626 


9.52436 


0.33447 


9.53495 


0.34273 


-44 41 


.50275 


.31824 


.51374 


.32640 


.52453 


.33461 


.53512 


.34287 


48 42 


.50294 


.31837 


.51393 


.32653 


.52471 


.33474 


.53530 


.34300 


52 43 


.50312 


.31851 


.51411 


.32667 


.52489 


.33488 


.53547 


.34314 


56 44 


.50331 


.31865 


.51429 


.32681 


.52507 


.33502 


.53565 


.34328 


8 ' 


4 A 35 m 68 


4* 30" 69 


4* 43" 70 


4* 47" 71 


45 


9.50349 


0.31878 


9.51447 


0.32694 


9.52525 


0.33515 


9.53582 


0.34342 


4 46 


.50368 


.31892 


.51465 


.32708 


.52542 


.33529 


.53600 


.34356 


5 47 


.50386 


.31905 


.51483 


.32721 


.52560 


.33543 


.53617 


.34369 


12 48 


.50405 


.31919 


.51501 


.32735 


.52578 


.33557 


.53635 


.34383 


16 49 


.50423 


.31932 


.51519 


.32749 


.52596 


.33570 


.53652 


.34397 


20 50 


9.50442 


0.31946 


9.51538 


0.32762 


9.52613 


0.33584 


9.53670 


0.34411 


24 51 


.50460 


.31959 


.51556 


.32776 


.52631 


.33598 


.53687 


.34425 


28 52 


.50478 


.31973 


.51574 


.32790 


.52649 


.33612 


.53704 


.34439 


32 53 


.50497 


.31987 


.51592 


.32803 


.52667 


.33625 


.53722 


.34452 


36 54 


.50515 


.32000 


.51610 


.32817 


.52684 


.33639 


.53739 


.34466 


^0 55 


9.50534 


0.32014 


9.51628 


0.32831 


9.52702 


0.33653 


9.53757 


0.34480 


44 56 


.50552 


.32027 


.51646 


.32844 


.52720 


.33667 


.53774 


.34494 


4S 57 


.50570 


.32041 


.51664 


.32858 


.52738 


.33680 


.53792 


.34508 


52 58 


.50589 


.32054 


.51682 


.32872 


.52755 


.33694 


.53809 


.34521 


56 59 


.50607 


.32068 


.51700 


.32885 


.52773 


.33708 


.53826 


.34535 


60 60 


9.50626 


0.32082 


9.51718 


0.32899 


9.52791 


0.33722 


9.53844 


0.34549 



268 



Table 10. Haversine Table 



s ' 


4^ 45 72 


4* 52 m 73 


4 h 56 m 74 


ffh O m 75 




Hav. 


No. 


Hav. 


No. 


Hav. 


No. 


Hav. 


No. 





9.53844 


0.34549 


9.54878 


0.35381 


9.55893 


0.36218 


9.56889 


0.37059 


4 1 


.53861 


.34563 


.54895 


.35395 


.55909 


.36232 


.56906 


.37073 


8 2 


.53879 


.34577 


.54912 


.35409 


.55926 


.36246 


.56922 


.37087 


12 3 


.53896 


.34591 


.54929 


.35423 


.55943 


.36260 


.56939 


.37101 


16 4 


.53913 


.34604 


.54946 


.35437 


.55960 


.36274 


.56955 


.37115 


20 5 


9.53931 


0.34618 


9.54963 


0.35451 


9.55976 


0.36288 


9.56972 


0.37129 


24 6 


.53948 


.34632 


.54980 


.35465 


.55993 


.36302 


.56988 


.37143 


28 7 


.53966 


.34646 


.54997 


.35479 


.56010 


.36316 


.57005 


.37157 


32 8 


.53983 


.34660 


.55014 


.35493 


.56027 


.36330 


.57021 


.37171 


36 9 


.54000 


.34674 


.55031 


.35507 


.56043 


.36344 


.57037 


.37186 


40 10 


9.54017 


0.34688 


9.55048 


0.35521 


9.56060 


0.36358 


9.57054 


0.37200 


44 11 


.54035 


.34701 


.55065 


.35534 


.56077 


.36372 


.57070 


.37214 


48 12 


.54052 


.34715 


.55082 


.35548 


.56093 


.36386 


.57087 


.37228 


52 13 


.54069 


.34729 


.55099 


.35562 


.56110 


.36400 


.57103 


.37242 


56 14 


.54087 


.34743 


.55116 


.35576 


.56127 


.36414 


.57119 


.37256 


s ' 


4*. 4,9"' 72 


4* 53 73 


4* 57 m 74 


gh lm 75 


15 


9.54104 


0.34757 


9.55133 


0.35590 


9.56144 


0.36428 


9.57136 


0.37270 


4 16 


.54121 


.34771 


.55150 


.35604 


.56160 


.36442 


.57152 


.37284 


5 17 


.54139 


.34784 


.55167 


.35618 


.56177 


.36456 


.57169 


.37298 


12 18 


.54156 


.34798 


.55184 


.35632 


.56194 


.36470 


.57185 


.37312 


iff 19 


.54173 


.34812 


.55201 


.35646 


.56210 


.36484 


.57201 


.37326 


20 20 


9.54190 


0.34826 


9.55218 


0.35660 


9.56227 


0.36498 


9.57218 


0.37340 


24 21 


.54208 


.34840 


.55235 


.35674 


.56244 


.36512 


.57234 


.37354 


25 22 


.54225 


.34854 


.55252 


.35688 


.56260 


.36526 


.57250 


.37368 


32 23 


.54242 


.34868 


.55269 


.35702 


.56277 


.36540 


.57267 


.37382 


Sff 24 


.54260 


.34882 


.55286 


.35716 


.56294 


.36554 


.57283 


.37397 


40 25 


9.54277 


0.34895 


9.55303 


0.35730 


9.56310 


0.36568 


9.57299 


0.37411 


44 26 


.54294 


.34909 


.55320 


.35743 


.56327 


.36582 


.57316 


.37425 


45 27 


.54311 


.34923 


.55337 


.35757 


.56343 


.36596 


.57332 


.37439 


52 28 


.54329 


.34937 


.55354 


.35771 


.56360 


.36610 


.57348 


.37453 


56 29 


.54346 


.34951 


.55370 


.35785 


.56377 


.36624 


.57365 


.37467 


s ' 


4* 50 m 72 


4 h 54 m 73 


4 A 58 74 


5 h 2 m 75 


30 


9.54363 


0.34965 


9.55387 


0.35799 


9.56393 


0.36638 


9.57381 


0.37481 


4 31 


.54380 


.34979 


.55404 


.35813 


.56410 


.36652 


.57397 


.37495 


8 32 


.54397 


.34992 


.55421 


.35827 


.56426 


.36666 


.57414 


.37509 


^2 33 


.54415 


.35006 


.55438 


.35841 


.56443 


.36680 


.57430 


.37523 


iff 34 


.54432 


.35020 


.55455 


.35855 


.56460 


.36694 


.57446 


.37537 


20 35 


9.54449 


0.35034 


9.55472 


0.35869 


9.56476 


0.36708 


9.57463 


0.37551 


24 36 


.54466 


.35048 


.55489 


.35883 


.56493 


.36722 


.57479 


.37566 


28 37 


.54483 


.35062 


.55506 


.35897 


.56509 


.36736 


.57495 


.37580 


32 38 


.54501 


.35076 


.55523 


.35911 


.56526 


.36750 


.57511 


.37594 


Sff 39 


.54518 


.35090 


.55539 


.35925 


.56543 


.36764 


.57528 


.37608 


40 40 


9.54535 


0.35103 


9.55556 


0.35939 


9.56559 


0.36778 


9.57544 


0.37622 


44 41 


.54552 


.35117 


.55573 


.35953 


.56576 


.36792 


.57560 


.37636 


48 42 


.54569 


.35131 


.55590 


.35967 


.56592 


.36806 


.57577 


.37650 


52 43 


.54587 


.35145 


.55607 


.35981 


.56609 


.36820 


.57593 


.37664 


50 44 


.54604 


.35159 


.55624 


.35995 


.56625 


.36834 


.57609 


.37678 


s ' 


4* 51 m 72 


4* 55 m 73 


4* 59 74 


5 h gm 75 


45 


9.54621 


0.35173 


9.55641 


0.36009 


9.56642 


0.36848 


9.57625 


0.37692 


4 46 


.54638 


.35187 


.55657 


.36023 


.56658 


.36862 


.57642 


.37706 


S 47 


.54655 


.35201 


.55674 


.36036 


.56675 


.36877 


.57658 


.37721 


.72 48 


.54672 


.35215 


.55691 


.36050 


.56692 


.36891 


.57674 


.37735 


16 49 


.54689 


.35228 


.55708 


.36064 


.56708 


.36905 


.57690 


.37749 


20 50 


9.54707 


0.35242 


9.55725 


0.36078 


9.56725 


0.36919 


9.57706 


0.37763 


24 51 


.54724 


.35256 


.55742 


.36092 


.56741 


.36933 


.57723 


.37777 


28 52 


.54741 


.35270 


.55758 


.36106 


.56758 


.36947 


.57739 


.37791 


32 53 


.54758 


.35284 


.55775 


.36120 


.56774 


.36961 


.57755 


.37805 


36 54 


.54775 


.35298 


.55792 


.36134 


.56791 


.36975 


.57771 


.37819 


40 55 


9.54792 


0.35312 


9.55809 


0.36148 


9.56807 


0.36989 


9.57787 


0.37833 


44 56 


.54809 


.35326 


.55826 


.36162 


.56824 


.37003 


.57804 


.37847 


48 57 


.54826 


.35340 


.55842 


.36176 


.56840 


.37017 


.57820 


.37862 


52 58 


.54843 


.35354 


.55859 


.36190 


.56856 


.37031 


.57836 


.37876 


56 59 


.54860 


.35368 


.55876 


.36204 


.56873 


.37045 


.57852 


.37890 


60 60 


9.54878 


0.35381 


9.55893 


0.36218 


9.56889 


0.37059 


9.57868 


0.37904 



Table 10. Haversine Table 



269 



s ' 


oh 4 m 76 


gh gm 77 


5 h 12 78 


fjh Iffn 79 




Hav. 


No. 


Hav. 


No. 


Hav. 


No. 


Hav. 


No. 





9.57868 


0.37904 


9.58830 


0.38752 


9.59774 


0.39604 


9.60702 


0.40460 


4 1 


.57885 


.37918 


.58846 


.38767 


.59790 


.39619 


.60717 


.40474 


8 2 


.57901 


.37932 


.58862 


.38781 


.59806 


.39633 


.60733 


.40488 


12 3 


.57917 


.37946 


.58878 


.38795 


.59821 


.39647 


.60748 


.40502 


16 4 


.57933 


.37960 


.58893 


.38809 


.59837 


.39661 


.60763 


.40517 


20 5 


9.57949 


0.37974 


9.58909 


0.38823 


9.59852 


0.39676 


9.60779 


0.40531 


24 6 


.57965 


.37989 


.58925 


.38837 


.59868 


.39690 


.60794 


.40545 


28 7 


.57981 


.38003 


.58941 


.38852 


.59883 


.39704 


.60809 


.40560 


32 8 


.57998 


.38017 


.58957 


.38866 


.59899 


.39718 


.60825 


.40574 


36 9 


.58014 


.38031 


.58973 


.38880 


.59915 


.39732 


.60840 


.40588 


40 10 


9.58030 


0.38045 


9.58989 


0.38894 


9.59930 


0.39746 


9.60855 


0.40602 


44 11 


.58046 


.38059 


.59004 


.38908 


.59946 


.39761 


.60870 


.40617 


48 12 


.58062 


.38073 


.59020 


.38923 


.59961 


.39775 


.60886 


.40631 


5 13 


.58078 


.38087 


.59036 


.38937 


.59977 


.39789 


.60901 


.40645 


56 14 


.58094 


.38102 


.59052 


.38951 


.59992 


.39803 


.60916 


.40660 


s ' 


5 h 5 m 76 


gh Qm 77 


5 h IS" 1 78 


5 h !Jm 790 


15 


9.58110 


0.38116 


9.59068 


0.38965 


9.60008 


0.39818 


9.60931 


0.40674 


4 16 


.58126 


.38130 


.59083 


.38979 


.60023 


.39832 


.60947 


.40688 


S 17 


.58143 


.38144 


.59099 


.38994 


.60039 


.39846 


.60962 


.40702 


12 18 


.58159 


.38158 


.59115 


.39008 


.60054 


.39861 


.60977 


.40717 


16 19 


.58175 


.38172 


.59131 


.39022 


.60070 


.39875 


.60992 


.40731 


20 20 


9.58191 


0.38186 


9.59147 


0.39036 


9.60085 


0.39889 


9.61008 


0.40745 


24 21 


.58207 


.38200 


.59162 


.39050 


.60101 


.39903 


.61023 


.40760 


..',s' 22 


.58223 


.38215 


.59178 


.39064 


.60116 


.39918 


.61038 


.40774 


32 23 


.58239 


.38229 


.59194 


.39079 


.60132 


.39932 


.61053 


.40788 


36 24 


.58255 


.38243 


.59210 


.39093 


.60147 


.39946 


.61069 


.40802 


40 25 


9.58271 


0.38257 


9.59225 


0.39107 


9.60163 


0.39960 


9.61084 


0.40817 


44 26 


.58287 


.38271 


.59241 


.39121 


.60178 


.39975 


.61099 


.40831 


45 27 


.58303 


.38285 


.59257 


.39135 


.60194 


.39989 


.61114 


.40845 


52 28 


.58319 


.38299 


.59273 


.39150 


.60209 


.40003 


.61129 


.40860 


56 29 


.58335 


.38314 


.59289 


.39164 


.60225 


.40017 


.61145 


.40874 


s ' 


5* 6 m 76 


5 h icr 77 


5ft 14 78 


5* IS" 1 79 


30 


9.58351 


0.38328 


9.59304 


0.39178 


9.60240 


0.40032 


9.61160 10.40888 


4 31 


.58367 


.38342 


.59320 


.39192 


.60256 


.40046 


.61175 


.40903 


5 32 


.58383 


.38356 


.59336 


.39206 


.60271 


.40060 


.61190 


.40917 


12 33 


.58399 


.38370 


.59351 


.39221 


.60287 


.40074 


.61205 


.40931 


16 34 


.58415 


.38384 


.59367 


.39235 


.60302 


.40089 


.61221 


.40945 


20 35 


9.58431 


0.38398 


9.59383 


0.39249 


9.60318 


0.40103 


9.61236 


0.40960 


24 36 


.58447 


.38413 


.59399 


.39263 


.60333 


.40117 


.61251 


.40974 


28 37 


.58463 


.38427 


.59414 


.39277 


.60348 


.40131 


.61266 


.40988 


32 38 


.58479 


.38441 


.59430 


.39292 


.60364 


.40146 


.61281 


.41003 


36 39 


.58495 


.38455 


.59446 


.39306 


.60379 


.40160 


.61296 


.41017 


40 40 


9.58511 


0.38469 


9.59461 


0.39320 


9.60395 


0.40174 


9.61312 


0.41031 


44 41 


.58527 


.38483 


.59477 


.39334 


.60410 


.40188 


.61327 


.41046 


48 42 


.58543 


.38498 


.59493 


.39348 


.60426 


.40203 


.61342 


.41060 


52 43 


.58559 


.38512 


.59508 


.39363 


.60441 


.40217 


.61357 


.41074 


56 44 


.58575 


.38526 


.59524 


.39377 


.60456 


.40231 


.61372 


.41089 


s ' 


5h jm 76 


5 h jj m 77 


5 h 15 m 78 


gh igm. 79 


6> 45 


9.58591 


0.38540 


9.59540 


0.39391 


9.60472 


0.40245 


9.61387 


0.41103 


4 46 


.58607 


.38554 


.59556 


.39405 


.60487 


.40260 


.61402 


.41117 


5 47 


.58623 


.38568 


.59571 


.39420 


.60502 


.40274 


.61417 


.41131 


12 48 


.58639 


.38582 


.59587 


.39434 


.60518 


.40288 


.61433 


.41146 


16 49 


.58655 


.38597 


.59602 


.39448 


.60533 


.40303 


.61448 


.41160 


20 50 


9.58671 


0.38611 


9.59618 


0.39462 


9.60549 


0.40317 


9.61463 


0.41174 


24 51 


.58687 


.38625 


.59634 


.39476 


.60564 


.40331 


.61478 


.41189 


28 52 


.58703 


.38639 


.59649 


.39491 


.60579 


.40345 


.61493 


.41203 


32 53 


.58719 


.38653 


.59665 


.39505 


.60595 


.40360 


.61508 


.41217 


36 54 


.58735 


.38667 


.59681 


.39519 


.60610 


.40374 


.61523 


.41232 


40 55 


9.58750 


0.38682 


9.59696 


0.39533 


9.60625 


0.40388 


9.61538 


0.41246 


44 56 


.58766 


.38696 


.59712 


.39548 


.60641 


.40402 


.61553 


.41260 


48 57 


.58782 


.38710 


.59728 


.39562 


.60656 


.40417 


.61568 


.41275 


52 58 


.58798 


.38724 


.59743 


.39576 


.60671 


.40431 


.61583 


.41289 


56 59 


.58814 


.38738 


.59759 


.39590 


.60687 


.40445 


.61598 


.41303 


60 60 


9.58830 


0.38752 


9.59774 


0.39604 


9.60702 


0.40460 


9.61614 


0.41318 



270 



Table 10. Haversine Table 



s ' 


5 h 20 80 


5* 24 m 81 


5h. 28 m 82 


O h 32'" 83 




Hav. 


No. 


Hav. 


No. 


Hav. 


No. 


Hav. 


No. 





9.61614 


0.41318 


9.62509 


0.42178 


9.63389 


0.43041 


9.64253 


0.43907 


4 1 


.61629 


.41332 


.62524 


.42193 


.63403 


.43056 


.64267 


.43921 


8 2 


.61644 


.41346 


.62538 


.42207 


.63418 


.43070 


.64281 


.43935 


12 3 


.61659 


.41361 


.62553 


.42221 


.63432 


.43085 


.64296 


.43950 


16 4 


.61674 


.41375 


.62568 


.42236 


.63447 


.43099 


.64310 


.43964 


20 5 


9.61689 


0.41389 


9.62583 


0.42250 


9.63461 


0.43113 


0.64324 


0.43979 


24 6 


.61704 


.41404 


.62598 


.42264 


.63476 


.43128 


.64339 


.43993 


28 7 


.61719 


.41418 


.62612 


.42279 


.63490 


.43142 


.64353 


.44008 


32 8 


.61734 


.41432 


.62627 


.42293 


.63505 


.43157 


.64367 


.44022 


36 9 


.61749 


.41447 


.62642 


.42308 


.63519 


.43171 


.64381 


.44036 


40 10 


9.61764 


0.41461 


9.62657 


0.42322 


9.63534 


0.43185 


9.64396 


0.44051 


44 11 


.61779 


.41475 


.62671 


.42336 


.63548 


.43200 


.64410 


.44065 


48 12 


.61794 


.41490 


.62686 


.42351 


.63563 


.43214 


.64424 


.44080 


52 13 


.61809 


.41504 


.62701 


.42365 


.63577 


.43229 


.64438 


.44094 


56 14 


.61824 


.41518 


.62716 


.42379 


.63592 


.43243 


.64452 


.44109 


s ' 


5h 21 m 80 


gh 25 m 81 


5 h 29 m 82 


gh 33 m 83 


15 


9.61839 


0.41533 


9.62730 


0.42394 


9.63606 


0.43257 


9.64467 


0.44123 


4 16 


.61854 


.41547 


.62745 


.42408 


.63621 


.43272 


.64481 


.44138 


S 17 


.61869 


.41561 


.62760 


.42423 


.63635 


.43286 


.64495 


.44152 


12 18 


.61884 


.41576 


.62774 


.42437 


.63649 


.43301 


.64509 


.44166 


J6 19 


.61899 


.41590 


.62789 


.42451 


.63664 


.43315 


.64523 


.44181 


20 20 


9.61914 


0.41604 


9.62804 


0.42466 


9.63678 


0.43330 


9.64538 


0.44195 


24 21 


.61929 


.41619 


.62819 


.42480 


.63693 


.43344 


.64552 


.44210 


25 22 


.61944 


.41633 


.62833 


.42494 


.63707 


.43358 


.64566 


.44224 


32 23 


.61959 


.41647 


.62848 


.42509 


.63722 


.43373 


.64580 


.44239 


30 24 


.61974 


.41662 


.62863 


.42523 


.63736 


.43387 


.64594 


.44253 


40 25 


9.61989 


0.41676 


9.62877 


0.42538 


9.63751 


0.43402 


9.64609 


0.44268 


44 26 


.62003 


.41690 


.62892 


.42552 


.63765 


.43416 


.64623 


.44282 


4S 27 


.62018 


.41705 


.62907 


.42566 


.63779 


.43430 


.64637 


.44296 


52 28 


.62033 


.41719 


.62921 


.42581 


.63794 


.43445 


.64651 


.44311 


56 29 


.62048 


.41733 


.62936 


.42595 


.63808 


.43459 


.64665 


.44325 


s ' 


5 h 22 80 


5* 26 m 81 


5 h 30 m 82 


5 h 34 m 83 


30 


9.62063 


0.41748 


9.62951 


0.42610 


9.63823 


0.43474 


9.64679 


0.44340 


4 31 


.62078 


.41762 


.62965 


.42624 


.63837 


.43488 


.64694 


.44354 


8 32 


.62093 


.41776 


.62980 


.42638 


.63851 


.43503 


.64708 


.44369 


12 33 


.62108 


.41791 


.62995 


.42653 


.63866 


.43517 


.64722 


.44383 


16 34 


.62123 


.41805 


.63009 


.42667 


.63880 


.43531 


.64736 


.44398 


20 35 


9.62138 


0.41819 


9.63024 


0.42681 


9.63895 


0.43546 


9.64750 


0.44412 


24 36 


.62153 


.41834 


.63039 


.42696 


.63909 


.43560 


.64764 


.44427 


28 37 


.62168 


.41848 


.63063 


.42710 


.63923 


.43575 


.64778 


.44441 


32 38 


.62182 


.41862 


.63068 


.42725 


.63938 


.43589 


.64793 


.44455' 


36 39 


.62197 


.41877 


.63082 


.42739 


.63952 


.43603 


.64807 


.44470 


40 40 


9.62212 


0.41891 


9.63097 


0.42753 


9.63966 


0.43618 


9.64821 


0.44484 


44 41 


.62227 


.41905 


.63112 


.42768 


.63981 


.43632 


.64835 


.44499 


48 42 


.62242 


.41920 


.63126 


.42782 


.63995 


.43647 


.64849 


.44513 


52 43 


.62257 


.41934 


.63141 


.42797 


.64010 


.43661 


.64863 


.44528 


56 44 


.62272 


.41949 


.63156 


.42811 


.64024 


.43676 


.64877 


.44542 


s ' 


5 h 23 >n 80 


5 h 27 m 81 


5>* 31 m 82 


5* S5 m 83 


45 


9.62287 


0.41963 


9.63170 


0.42825 


9.64038 


0.43690 


9.64891 


0.44557 


4 46 


.62301 


.41977 


.63185 


.42840 


.64053 


.43704 


.64905 


.44571 


5 47 


.62316 


.41992 


.63199 


.42854 


.64067 


.43719 


.64919 


.44586 


12 48 


.62331 


.42006 


.63214 


.42869 


.64081 


.43733 


.64934 


.44600 


16 49 


.62346 


.42020 


.63228 


.42883 


.64096 


.43748 


.64948 


.44614 


20 50 


9.62361 


0.42035 


9.63243 


0.42897 


9.64110 


0.43762 


9.64962 


0.44629 


24 51 


.62376 


.42049 


.63258 


.42912 


.64124 


.43777 


.64976 


.44643 


> V KO 

.< i OZ 


.62390 


.42063 


.63272 


.42926 


.64139 


.43791 


.64990 


.44658 


32 53 


.62405 


.42078 


.63287 


.42941 


.64153 


.43805 


.65004 


.44672 


36 54 


.62420 


.42092 


.63301 


.42955 


.64167 


.43820 


.65018 


.44687 


40 55 


9.62435 


0.42106 


9.63316 


0.42969 


9.64181 


0.43834 


9.65032 


0.44701 


44 56 


.62450 


.42121 


.63330 


.42984 


.64196 


.43849 


.65046 


.44716 


48 57 


.62464 


.42135 


.63345 


.42998 


.64210 


.43863 


.65060 


.44730 


52 58 


.62479 


.42150 


.63360 


.43013 


.64224 


.43878 


.65074 


.44745 


56 59 


.62494 


.42164 


.63374 


.43027 


.64239 


.43892 


.65088 


.44759 


60 60 


9.62509 


0.42178 


9.63389 


0.43041 


9.64253 


0.43907 


9.65102 


0.44774 



Table 10. Haversine Table 



271 



s ' 


5* 36'" 84 


5* 40 m 85 


5A 44 m 86 


5* 48 m 87" 




Hav. 


No. 


Hav. 


No. 


Hav. 


No. 


Hav. 


No. 





9.65102 


0.44774 


9.65937 


0.45642 


9.66757 


0.46512 


9.67562 


0.47383 


4 1 


.65116 


.44788 


.65950 


.45657 


.66770 


.46527 


.67576 


.47398 


8 2 


.65130 


.44803 


.65964 


.45671 


.66784 


.46541 


.67589 


.47412 


12 3 


.65144 


.44817 


.65978 


.45686 


.66797 


.46556 


.67602 


.47427 


16 4 


.65158 


.44831 


.65992 


.45700 


.66811 


.46570 


.67616 


.47441 


20 5 


9.65172 


0.44846 


9.66006 


0.45715 


9.66824 


0.46585 


9.67629 


0.47456 


24 6 


.65186 


.44860 


.66019 


.45729 


.66838 


.46599 


.67642 


.47470 


28 7 


.65200 


.44875 


.66033 


.45744 


.66851 


.46614 


.67656 


.47485 


32 8 


.65214 


.44889 


.66047 


.45758 


.66865 


.46628 


.67669 


.47499 


36 9 


.65228 


.44904 


.66061 


.45773 


.66878 


.46643 


.67682 


.47514 


40 10 


9.65242 


0.44918 


9.66074 


0.45787 


9.66892 


0.46657 


9.67695 


0.47528 


44 11 


.65256 


.44933 


.66088 


.45802 


.66905 


.46672 


.67709 


.47543 


48 12 


.65270 


.44947 


.66102 


.45816 


.66919 


.46686 


.67722 


.47558 


52 13 


.65284 


.44962 


.66116 


.45831 


.66932 


.46701 


.67735 


.47572 


56 14 


.65298 


.44976 


.66129 


.45845 


.66946 


.46715 


.07748 


.47587 


s ' 


5 h 37 m 84 


5* 41 m 85 


5 h 45 m 86 


,-;'< 4<> m 87 


15 


9.65312 


0.44991 


9.66143 


0.45860 


9.66959 


0.46730 


9.67762 


0.47601 


4 16 


.65326 


.45005 


.66157 


.45874 


.66973 


.46744 


.67775 


.47616 


S 17 


.65340 


.45020 


.66170 


.45889 


.66986 


.46759 


.67788 


.47630 


12 18 


.65354 


.45034 


.66184 


.45903 


.67000 


.46773 


.67801 


.47645 


16 19 


.65368 


.45048 


.66198 


.45918 


.67013 


.46788 


.67815 


.47659 


20 20 


9.65382 


0.45063 


9.66212 


0.45932 


9.67027 


0.46802 


9.67828 


0.47674 


24 21 


.65396 


.45077 


.66225 


.45947 


.67040 


.46817 


.67841 


.47688 


25 22 


.65410 


.45092 


.66239 


.45961 


.67054 


.46831 


.67854 


.47703 


32 23 


.65424 


.45106 


.66253 


.45976 


.67067 


.46846 


.67868 


.47717 


30 24 


.65438 


.45121 


.66266 


.45990 


.67081 


.46860 


.67881 


.47732 


40 25 


9.65452 


0.45135 


9.66280 


0.46005 


9.67094 


0.46875 


9.67894 


0.47746 


44 26 


.65466 


.45150 


.66294 


.46019 


.67108 


.46890 


.67907 


.47761 


45 27 


.65480 


.45164 


.66307 


.46034 


.67121 


.46904 


.67920 


.47775 


52 28 


.65493 


.45179 


.66321 


.46048 


.67134 


.46919 


.67934 


.47790 


56 29 


.65507 


.45193 


.66335 


.46063 


.67148 


.46933 


.67947 


.47805 


s ' 


5* 38 m 84 


5* 42 85 


5 h 4(>'" 86 


') h 50 m 87 


30 


9.65521 


0.45208 


9.66348 


0.46077 


9.67161 


0.46948 


V). 67960 


0.47819 


4 31 


.65535 


.45222 


.66362 


.46092 


.67175 


.46962 


.67973 


.47834 


8 32 


.65549 


.45237 


.66376 


.46106 


.67188 


.46977 


.67986 


.47848 


J2 33 


.65563 


.45251 


.66389 


.46121 


.67202 


.46991 


.68000 


.47863 


/6 34 


.65577 


.45266 


.66403 


.46135 


.67215 


.47006 


.68013 


.47877 


20 35 


9.65591 


0.45280 


9.66417 


0.46150 


9.67228 


0.47020 


9.68026 


0.47892 


24 36 


.65605 


.45295 


.66430 


.46164 


.67242 


.47035 


.68039 


.47906 


28 37 


.65619 


.45309 


.66444 


.46179 


.67255 


.4704& 


.68052 


.47921 


32 38 


.65632 


.45324 


.66458 


.46193 


.67269 


.47064 


.68066 


.47935 


36 39 


.65646 


.45338 


.66471 


.46208 


.67282 


.47078 


.68079 


.47950 


40 40 


9.65660 


0.45353 


9.66485 


0.46222 


9.67295 


0.47093 


9.68092 


0.47964 


44 41 


.65674 


.45367 


.66499 


.46237 


.67309 


.47107 


.68105 


.47979 


48 42 


.65688 


.45381 


.66512 


.46251 


.67322 


.47122 


.68118 


.47993 


52 43 


.65702 


.45396 


.66526 


.46266 


.67336 


.47136 


.68131 


.48008 


50 44 


.65716 


.45410 


.66539 


.46280 


.67349 


.47151 


.68144 


.48022 


s ' 


5h 39 84 


5* 43 m 85 


5 h 47 m 86" 


6* 51 m 87 


45 


9.65729 


0.45425 


9.66553 


0.46295 


9.67362 


0.47165 


9.68158 


0.48037 


4 46 


.65743 


.45439 


.66567 


.46309 


.67376 


.47180 


.68171 


.48052 


5 47 


.65757 


.45454 


.66580 


.46324 


.67389 


.47194 


.68184 


.48066 


J2 48 


.65771 


.45468 


.66594 


.46338 


.67402 


.47209 


.68197 


.48081 


16 49 


.65785 


.45483 


.66607 


.46353 


.67416 


.47223 


.68210 


.48095 


20 50 


9.65799 


0.45497 


9.66621 


0.46367 


9.67429 


0.47238 


9.68223 


0.48110 


24 51 


.65812 


.45512 


.66635 


.46382 


.67443 


.47252 


.68236 


.48124 


28 52 


.65826 


.45526 


.66648 


.46396 


-.67456 


.47267 


.68249 


.48139 


32 53 


.65840 


.45541 


.66662 


.46411 


.67469 


.47282 


.68263 


.48153 


30 54 


.65854 


.45555 


.66675 


.46425 


.67483 


.47296 


.68276 


.48168 


40 55 


9.65868 


0.45570 


9.66689 


0.46440 


9.67496 


0.47311 


9.68289 


0.48182 


44 56 


.65881 


.45584 


.66702 


.46454 


.67509 


.47325 


.68302 


.48197 


48 57 


.65895 


.45599 


.66716 


.46469 


.67522 


.47340 


.68315 


.48211 


52 58 


.65909 


.45613 


.66730 


.46483 


.67536 


.47354 


.68328 


.48226 


56 59 


.65923 


.45628 


.66743 


.46498 


.67549 


.47369 


.68341 


.48241 


60 60 


9.65937 


0.45642 


9.66757 


0.46512 


9.67562 


0.47383 


9.68354 


0.48255 



272 



Table 10. Haversine Table 



s ' 


5 h 52 m 88 


5* 56 89 






Qh Q 6* 4 m 




Hav. 


No. 


Hav. 


No. 






Hav. 


Hav. 





9.68354 


0.48255 


9.69132 


0.49127 







9.69897 


9.70648 


4 1 


.68367 


.48269 


.69145 


.49142 




4 


.69910 


.70661 


5 2 


.68380 


.48284 


.69158 


.49156 




8 


.69922 


.70673 


12 3 


.68393 


.48299 


.69171 


.49171 




12 


.69935 


.70686 


16 4 


.68407 


.48313 


.69184 


.49186 




16 


.69948 


.70698 


20 5 


9.68420 


0.48328 


9.69197 


0.49200 




20 


9.69960 


9.70710 


24 6 


.68433 


.48342 


.69209 


.49215 




24 


.69973 


.70723 


28 1 


.68446 


.48357 


.69222 


.49229 




28 


.69985 


.70735 


32 8 


.68459 


.48371 


.69235 


.49244 




32 


.69998 


.70748 


36 9 


.68472 


.48386 


.69248 


.49258 




36 


.70011 


.70760 


40 10 


9.68485 


0.48400 


9.69261 


0.49273 




40 


9.70023 


9.70772 


44 11 


.68498 


.48415 


.69274 


.49287 




44 


.70036 


.70785 


48 12 


.68511 


.48429 


.69286 


.49302 




48 


.70048 


.70797 


T5> 1 5 
' . J-O 


.68524 


.48444 


.69299 


.49316 


TO 

5 


52 


.70061 


.70809 


56 14 


.68537 


.48459 


.69312 


.49331 




56 


.70074 


.70822 


s ' 


5 h 53 88 


5h 57> 89 



9 


s 


Qh jm Qh gm 


15 


9.68550 


0.48473 


9.69325 


0.49346 


i 





9.70086 


9.70834 


4 16 


.68563 


.48488 


.69338 


.49360 


M 


4 


.70099 


.70847 


8 17 


.68576 


.48502 


.69350 


.49375 


6 


8 


.70111 


.70859 


/2 18 


.68589 


.48517 


.69363 


.49389 


fe 


12 


.70124 


.70871 


16 19 


.68602 


.48531 


.69376 


.49404 


1 


16 


.70136 


.70884 


20 20 


9.68615 


0.48546 


9.69389 


0.49418 




20 


9.70149 


9.70896 


24 21 


.68628 


.48560 


.69402 


.49433 


32 

CJ 


24 


.70161 


.70908 


28 22 


.68641 


.48575 


.69414 


.49447 




28 


.70174 


.70921 


32 23 


.68654 


.48589 


.69427 


.49462 


2' 


32 


.70187 


.70933 


36 24 


.68667 


.48604 


.69440 


.49476 


5> 


36 


.70199 


.70945 


40 25 


9.68680 


0.48618 


9.69453 


0.49491 


.2 ** 


40 


9.70212 


9.70958 


/ / 9 fi 


.68693 


.48633 


.69465 


.49506 


"5* 1 


44 


.70224 


.70970 


4<S 27 


.68706 


.48648 


.69478 


.49520 


^ 


48 


.70237 


.70982 


52 28 


.68719 


.48662 


.69491 


.49535 


2-s 


52 


.70249 


.70995 


56 29 


.68732 


.48677 


.69504 


.49549 


x, O 
O O 


56 


.70262 


.71007 


s ' 


5* 54 m 88 


5* 58" 1 89 


*-" >> 


s 


Qh #m Qh Qm 


30 


9.68745 


0.48691 


9.69516 


0.49564 


"5 





9.70274 


9.71019 


4 31 


.68758 


.48706 


.69529 


.49578 


.S4j 


4 


.70287 


.71032 


.V 32 


.68771 


.48720 


.69542 


.49593 




8 


.70299 


.71044 


i - 33 


.68784 


.48735 


.69555 


.49607 


o Q 


12 


.70312 


.71056 


^6 34 


.68797 


.48749 


.69567 


.49622 


.? *> 


16 


.70324 


.71068 


20 35 


9.68810 


0.48764 


9.69580 


0.49636 


S 


20 


9.70337 


9.71081 


24 36 


.68823 


.48778 


.69593 


.49651 





24 


.70349 


.71093 


28 37 


.68836 


.48793 


.69605 


.49665 


'" 05 


28 


.70362 


.71105 


' .' OO 


.68849 


.48807 


.69618 


.49680 


d 

fl 


32 


.70374 


.71118 


&? Oft 

>' > *>9 


.68862 


.48822 


.69631 


.49695 


Q 


36 


.70387 


.71130 


40 40 


9.68875 


0.48837 


9.69644 


0.49709 


"o 


40 


9.70399 


9.71142 


44 41 


.68887 


.48851 


.69656 


.49724 




44 


.70412 


.71154 


48 42 


.68900 


.48866 


.69669 


.49738 


e 


48 


.70424 


.71167 


52 43 


.68913 


.48880 


.69682 


.49753 




52 


.70437 


.71179 


oo 44 


.68926 


.48895 


.69694 


.49767 




56 


.70449 


.71191 


s ' 


5>> 55 m 88 


5* 59 m 89 


1 


s 


Qh 3>n Qh 7 


45 


9.68939 


0.48909 


9.69707 


0.49782 


1 





9.70462 


9.71203 


4 46 


.68952 


.48924 


.69720 


.49796 


3 


4 


.70474 


.71216 


S 47 


.68965 


.48938 


.69732 


.49811 


o 


8 


.70487 


.71228 


12 48 


.68978 


.48953 


.69745 


.49825 


K 


12 


.70499 


.71240 


/6 49 


.68991 


.48967 


.69758 


.49840 




16 


.70512 


.71252 


20 50 


9.69004 


0.48982 


9.69770 


0.49855 




20 


9.70524 


9.71265 


24 51 


.69017 


.48997 


.69783 


.49869 




24 


.70537 


.71277 


28 52 


.69029 


.49011 


.69796 


.49884 




28 


.70549 


.71289 


> CO 
' - OO 


.69042 


.49026 


.69808 


.49898 




32 


.70561 


.71301 


36 54 


.69055 


.49040 


.69821 


.49913 




36 


.70574 


.71314 


40 55 


9.69068 


0.49055 


9.69834 


0.49927 




40 


9.70586 


9.71326 


44 56 


.69081 


.49069 


.69846 


.49942 




44 


.70599 


.71338 


48 57 


.69094 


.49084 


.69859 


.49956 




48 


.70611 


.71350 


52 58 


.69107 


.49098 


.69872 


.49971 




52 


.70624 


.71362 


56 59 


.69120 


.49113 


.69884 


.49985 




56 


.70636 


.71375 


60 60 


9.69132 


0.49127 


9.69897 


0.50000 




60 


9.70648 


9.71387 



Table 10. Haversine Table 



273 



s 


Qh 8 Qh J^m 


Qh IQm Qh 20 


Qh 24 Qh 28 


Qh S2 6+ 86 




Hav. 


Hav. 


Hav. 


Hav. 


Hav. 


Hav. 


Hav. 


Hav. 





9.71387 


9.72112 


9.72825 


9.73526 


9.74215 


9.74891 


9.75556 


9.76209 


4 


.71399 


.72124 


.72837 


.73538 


.74226 


.74902 


.75567 


.76220 


8 


.71411 


.72136 


.72849 


.73549 


.74237 


.74914 


.75578 


.76231 


12 


.71423 


.72148 


.72861 


.73561 


.74249 


.74925 


.75589 


.76241 


16 


.71436 


.72160 


.72873 


.73572 


.74260 


.74936 


.75600 


.76252 


20 


9.71448 


9.72172 


9.72884 


9.73584 


9.74272 


9.74947 


9.75611 


9.76263 


24 


.71460 


.72184 


.72896 


.73596 


.74283 


.74958 


.75622 


.76274 


28 


.71472 


.72196 


.72908 


.73607 


.74294 


.74969 


.75633 


.76285 


32 


.71484 


.72208 


.72920 


.73619 


.74306 


.74981 


.75644 


.76296 


36 


.71496 


.72220 


.72931 


.73630 


.74317 


.74992 


.75655 


.76306 


40 


9.71509 


9.72232 


9.72943 


9.73642 


9.74328 


9.75003 


9.75666 


9.76317 


44 


.71521 


.7234 


.72955 


.73653 


.74340 


.75014 


.75677 


.76328 


48 


.71533 


.72256 


.72967 


.73665 


.74351 


.75025 


.75688 


.76338 


52 


.71545 


.72268 


.72978 


.73676 


.74362 


.75036 


.75698 


.76349 


56 


.71557 


.72280 


.72990 


.73688 


.74374 


.75047 


.75709 


.76360 


s 


6 h 9 6 h IS 


Qh Ijm Q h 2im 


Qh 25 6* 29 


Qh sgm Qh 37 m 





9.71569 


9.72292 


9.73002 


9.73699 


9.74385 


9.75059 


9.75720 


9.76371 


4 


.71582 


.72304 


.73014 


.73711 


.74396 


.75070 


.75731 


.76381 


8 


.71594 


.72316 


.73025 


.73722 


.74408 


.75081 


.75742 


.76392 


12 


.71606 


.72328 


.73037 


.73734 


.74419 


.75092 


.75753 


.76403 


16 


.71618 


.72340 


.73049 


.73746 


.74430 


.75103 


.75764 


.76414 


20 


9.71630 


9.72352 


9.73060 


9.73757 


9.74442 


9.75114 


9.75775 


9.76424 


24 


.71642 


.72363 


.73072 


.73769 


.74453 


.75125 


.75786 


.76435 


28 


.71654 


.72375 


.73084 


.73780 


.74464 


.75136 


.75797 


.76446 


32 


.71666 


.72387 


.73096 


.73792 


.74475 


.75147 


.75808 


.76456 


36 


.71679 


.72399 


.73107 


.73803 


.74487 


.75159 


.75819 


.76467 


40 


9.71691 


9.72411 


9.73119 


9.73815 


9.74498 


9.75170 


9.75830 


9.76478 


44 


.71703 


.72423 


.73131 


.73826 


.74509 


.75181 


.75840 


.76489 


48 


.71715 


.72435 


.73142 


.73838 


.74521 


.75192 


.75851 


.76499 


52 


.71727 


.72447 


.73154 


.73849 


.74532 


.75203 


.75862 


.76510 


56 


.71739 


.72459 


.73166 


.73860 


.74543 


.75214 


.75873 


.76521 


8 


6 h 10 6* 14 


Qh is e* 22 


6* 26 6 h SO 


6 h 34 6 h 38 





9.71751 


9.72471 


9.73177 


9.73872 


9.74554 


9.75225 


9.75884 


9.76531 


4 


.71763 


.72482 


.73189 


.73883 


.74566 


.75236 


.75895 


.76542 


8 


.71775 


.72494 


.73201 


.73895 


.74577 


.75247 


.75906 


.76553 


12 


.71787 


.72506 


.73212 


.73906 


.74588 


.75258 


.75917 


.76563 


16 


.71800 


.72518 


.73224 


.73918 


.74600 


.75269 


.75927 


.76574 


20 


9.71812 


9.72530 


9.73236 


9.73929 


9.74611 


9.75280 


9.75938 


9.76585 


24 


.71824 


.72542 


.73247 


.73941 


.74622 


.75291 


.75949 


.76595 


28 


.71836 


.72554 


.73259 


.73952 


.74633 


.75303 


.75960 


.76606 


32 


.71848 


.72565 


.73271 


.73964 


.74645 


.75314 


.75971 


.76617 


36 


.71860 


.72577 


.73282 


.73975 


.74656 


.75325 


.75982 


.76627 


40 


9.71872 


9.72589 


9.73294 


9.73987 


9.74667 


9.75336 


9.75993 


9.76638 


44 


.71884 


.72601 


.73306 


.73998 


.74678 


.75347 


.76004 


.76649 


48 


.71896 


.72613 


.73317 


.74009 


.74690 


.75358 


.76014 


.76659 


52 


.71908 


.72625 


.73329 


.74021 


.74701 


.75369 


.76025 


.76670 


56 


.71920 


.72637 


.73341 


.74032 


.74712 


.75380 


.76036 


.76681 


s 


Qh jjm 6* 15 m 


Qh igm Qh 23 


Qh 27 6* 31 


Qh 35m Qh 39 m 





9.71932 


9.72648 


9.73352 


9.74044 


9.74723 


9.75391 


9.76047 


9.76691 


4 


.71944 


.72660 


.73364 


.74055 


.74734 


.75402 


.76058 


.76702 


8 


.71956 


.72672 


.73375 


.74067 


.74746 


.75413 


.76069 


.76713 


12 


.71968 


.72684 


.73387 


.74078 


.74757 


.75424 


.76079 


.76723 


16 


.71980 


.72696 


.73399 


.74089 


.74768 


.75435 


.76090 


.76734 


20 


9.71992 


9.72708 


9.73410 


9.74101 


9.74779 


9.75446 


9.76101 


9.76745 


24 


.72004 


.72719 


.73422 


.74112 


.74791 


.75457 


.76112 


.76755 


28 


.72016 


.72731 


.73433 


.74124 


.74802 


.75468 


.76123 


.76766 


32 


.72028 


.72743 


.73445 


.74135 


.74813 


.75479 


.76134 


.76777 


36 


.72040 


.72755 


.73457 


.74146 


.74824 


.75490 


.76144 


.76787 


40 


9.72052 


9.72767 


9.73468 


9.74158 


9.74835 


9.75501 


9.76155 


9.76798 


44 


.72064 


.72778 


.73480 


.74169 


.74846 


.75512 


.76166 


.76808 


48 


.72076 


.72790 


.73491 


.74181 


.74858 


.75523 


.76177 


.76819 


52 


.72088 


.72802 


.73503 


.74192 


.74869 


.75534 


.76188 


.76830 


56 


.72100 


.72814 


.73515 


.74203 


.74880 


.75545 


.76198 


.76840 


60 


9.72112 


9.72825 


9.73526 


9.74215 


9.74891 


9.75556 


9.76209 


9.76851 



274 



Table 10. Haversine Table 



s 


eh 4o m ti* 44 m 


0* 48 6>> 52 m 


6 h 56 7 h O m 


7 h 4 m 7 h 8 




Hav. 


Hav. 


Hav. 


Hav. 


Hav. 


Hav. 


Hav. 


Hav. 





9.76851 


9.77481 


9.78101 


9.78709 


9.79306 


9.79893 


9.8U470 


9.81036 


4 


.76861 


.77492 


.78111 


.78719 


.79316 


.79903 


.80479 


.81045 


8 


.76872 


.77502 


.78121 


.78729 


.79326 


.79913 


.80489 


.81054 


12 


.76883 


.77512 


.78131 


.78739 


.79336 


.79922 


.80498 


.81064 


16 


.76893 


.77523 


.78141 


.78749 


.79346 


.79932 


.80508 


.81073 


20 


9.76904 


9.77533 


9.78152 


9.78759 


9.79356 


9.79942 


9.80517 


9.81082 


94 


.76914 


.77544 


.78162 


.78769 


.79366 


.79951 


.80527 


.81092 


28 


.76925 


.77554 


.78172 


.78779 


.79376 


.79961 


.80536 


.81101 


32 


.76936 


.77564 


.78182 


.78789 


.79385 


.79971 


.80546 


.81110 


36 


.76946 


.77575 


.78192 


.78799 


.79395 


.79980 


.80555 


.81120 


40 


9.76957 


9.77585 


9.78203 


9.78809 


9.79405 


9.79990 


9.80565 


9.88129 


44 


.76967 


.77596 


.78213 


.78819 


.79415 


.80000 


.80574 


.81138 


48 


.76978 


.77606 


.78223 


.78829 


.79425 


.80009 


.80584 


.81148 


52 


.76988 


.77616 


.78233 


.78839 


.79434 


.80019 


.80593 


.81157 


56 


.76999 


.77627 


.78243 


.78849 


.79444 


.80029 


.80603 


.81166 


s 


Qh ^m Qh ^ 5 m 


6* 49 6* 53 


6 h 57 m 7 h l m 


7 h 5 m 7 h 9 m 





9.77009 


9.77637 


9.78254 


9.78859 


9.79454 


9.8003S 


9.80612 


9.81176 


4 


.77020 


.77647 


.78264 


.78869 


.79464 


.80048 


.80622 


.81185 


8 


.77031 


.77658 


.78274 


.78879 


.79474 


.80058 


.80631 


.81194 


12 


.77041 


.77668 


.78284 


.78889 


.79484 


.80067 


.80641 


.81204 


16 


.77052 


.77679 


.78294 


.78899 


.79493 


.80077 


.80650 


.81213 


20 


9.77062 


9.77689 


9.78305 


9.78909 


9.79503 


9.80087 


9.80660 


9.81222 


24 


.77073 


.77699 


.78315 


.78919 


.79513 


.80096 


.80669 


.81231 


28 


.77083 


.77710 


.78325 


.78929 


.79523 


.80106 


.80678 


.81241 


32 


.77094 


.77720 


.78335 


.78939 


.79533 


.80116 


.80688 


.81250 


36 


.77104 


.77730 


.78345 


.78949 


.79542 


.80125 


.80697 


.81259 


40 


9.77115 


9.77741 


9.78355 


9.78959 


9.79552 


9.80135 


9.80707 


9.81269 


44 


.77125 


.77751 


.78365 


.78969 


.79562 


.80144 


.80716 


.81278 


48 


.77136 


.77761 


.78376 


.78979 


.79572 


.80154 


.80726 


.81287 


52 


.77146 


.77772 ' 


.78386 


.78989 


.79582 


.80164 


.80735 


.81296 


56 


.77157 


.77782 


.78396 


.78999 


.79591 


.80173 


.80745 


.81306 


s 


#> 42 6 h 46 


Qh go m 6 h 64 m 


Qh 58 7 h 2 m 


7A Qm 7 h lQ m 





9.77167 


9.77792 


9.78406 


9.79009 


9.79601 


9.80183 


9.80754 


9.81315 


4 


.77178 


.77803 


.78416 


.79019 


.79611 


.80192 


.80763 


.81324 


8 


.77188 


.77813 


.78426 


.79029 


.79621 


.80202 


.80773 


.81333 


12 


.77199 


.77823 


.78436 


.79039 


.79631 


.80212 


.80782 


.81343 


16 


.77209 


.77834 


.78447 


.79049 


.79640 


.80221 


.80792 


.81352 


20 


9.77220 


9.77844 


9.78457 


9.79059 


9.79650 


9.80231 


9.80801 


9.81361 


24 


.77230 


.77854 


.78467 


.79069 


.79660 


.80240 


.80811 


.81370 


28 


.77241 


.77864 


.78477 


.79079 


.79670 


.80250 


.80820 


.81380 


32 


.77251 


.77875 


.78487 


.79089 


.79679 


.80260 


.80829 


.81389 


36 


.77262 


.77885 


.78497 


.79099 


.79689 


.80269 


.80839 


.81398 


40 


9.77272 


9.77895 


9.78507 


9.79108 


9.79699 


9.80279 


9.80848 


9.81407 


44 


.77283 


.77906 


.78517 


.79118 


.79709 


.80288 


.80858 


.81417 


48 


.77293 


.77916 


.78528 


.79128 


.79718 


.80298 


.80867 


.81426 


52 


.77304 


.77926 


.78538 


.79138 


.79728 


.80307 


.80876 


.81435 


56 


.77314 


.77936 


.78548 


.79148 


.79738 


.80317 


.30886 


.81444 


s 


6h 43*1 Qh tfrn 


Qh Qim Qh ggm 


Qh 5Qm Jh gni 


7 A 7< 7A jjm 





9.77325 


9.77947 


9.78558 


9.79158 


9.79748 


9.80327 


9.80895 


9.81454 


4 


.77335 


.77957 


.78568 


.79168 


.79757 


.80336 


.80905 


.81463 


8 


.77346 


.77967 


.78578 


.79178 


.79767 


.80346 


.80914 


.81472 


1-2 


.77356 


.77978 


.78588 


.79188 


.79777 


.80355 


.80923 


.81481 


16 


.77366 


.77988 


.78598 


.79198 


.79787 


.80365 


.80933 


.81490 


20 


9.77377 


9.77998 


9.78608 


9.79208 


9.79796 


9.80374 


9.80942 


9.81500 


24 


.77387 


.78008 


.78618 


.79217 


.79806 


.80384 


.80952 


.81509 


28 


.77398 


.78019 


.78628 


.79227 


.79816 


.80393 


.80961 


.81518 


32 


.77408 


.78029 


.78638 


.79237 


.79825 


.80403 


.80970 


.81527 


36 


.77419 


.78039 


.78649 


.79247 


.79835 


.80413 


.80980 


.81536 


40 


9.77429 


9.78049 


9.78659 


9.79257 


9.79845 


9.80422 


9.80989 


9.81546 


44 


.77440 


.78060 


.78669 


.79267 


.79855 


.80432 


.80998 


.81555 


48 


.77450 


.78070 


.78679 


.79277 


.79864 


.80441 


.81008 


.81564 


52 


.77460 


.78080 


.78689 


.79287 


.79874 


.80451 


.81017 


.81573 


56 


.77471 


.78090 


.78699 


.79297 


.79884 


.80460 


.81026 


.81582 


60 


9.77481 


9.78101 


9.78709 


9.79306 


9.79893 


9.80470 


9.81036 


9.81592 



Table 10. Haversine Table 



275 



s 


7 h Igm 7 h 16 m 


7 h 20 m 7 h 24 m 


7 A 28 m 7 h 32 


7 h 3Qm 7 h Jflm 




Hav. 


Hav. 


Hav. 


Hav. 


Hav. 


Hav. 


Hav. 


Hav. 





9.81592 


9.82137 


9.82673 


9.83199 


9.83715 


9.84221 


9.84718 


9.85206 


4 


.81601 


.82146 


.82682 


.83207 


.83723 


.84230 


.84726 


.85214 


8 


.81610 


.82155 


.82691 


.83216 


.83732 


.84238 


.84735 


.85222 


12 


.81619 


.82164 


.82699 


.83225 


.83740 


.84246 


.84743 


.85230 


16 


.81628 


.82173 


.82708 


.83233 


.83749 


.84255 


.84751 


.85238 


W 


9.81637 


9.82182 


9.82717 


9.83242 


9.83757 


9.84263 


9.84759 


9.85246 


24 


.81647 


.82191 


.82726 


.83251 


.83766 


.84271 


.84767 


.85254 


28 


.81656 


.82200 


.82735 


.83259 


.83774 


.84280 


.84776 


.85262 


32 


.81665 


.82209 


.82744 


.83268 


.83783 


.84288 


.84784 


.85270 


36 


.81674 


.82218 


.82752 


.83277 


.83791 


.84296 


.84792 


.85278 


40 


9.81683 


9.82227 


9.82761 


9.83285 


9.83800 


9.84305 


9.84800 


9.85286 


44 


.81692 


.82236 


.82770 


.83294 


.83808 


.84313 


.84808 


.85294 


48 


.81701 


.82245 


.82779 


.83303 


.83817 


.84321 


.84817 


.85302 


52 


.81711 


.82254 


.82788 


.83311 


.83825 


.84330 


.84825 


.85310 


56 


.81720 


.82263 


.82796 


.83320 


.83834 


.84338 


.84833 


.85318 


s 


7* 13 m 7* 17 m 


7 h 21 7* 25 m 


7 h 29 7* 33 m 


7 h 37 m 7 h ^m 





9.81729 


9.82272 


9.82805 


9.83329 


9.83842 


9.84346 


9.S4841 


9.85326 


4 


.81738 


.82281 


.82814 


.83337 


.83851 


.84355 


.84849 


.85334 


8 


.81747 


.82290 


.82823 


.83346 


.83859 


.84363 


.84857 


.85342 


12 


.81756 


.82299 


.82832 


.83355 


.83868 


.84371 


.84866 


.85350 


16 


.81765 


.82308 


.82840 


.83363 


.83876 


.84380 


.84874 


.85358 


20 


9.81775 


9.82317 


9.82849 


9.83372 


9.83885 


9.84388 


9.84882 


9.85366 


24 


.81784 


.82326 


.82858 


.83380 


.83893 


.84396 


.84890 


.85374 


28 


.81793 


.82335 


.82867 


.83389 


.83902 


.84405 


.84898 


.85382 


32 


.81802 


.82344 


.82876 


.83398 


.83910 


.84413 


.84906 


.85390 


36 


.81811 


.82353 


.82884 


.83406 


.83919 


.84421 


.84914 


.85398 


40 


9.81820 


9.82362 


9.82893 


9.83415 


9.83927 


9.84430 


9.84923 


9.85406 


44 


.81829 


.82371 


.82902 


.83424 


.83935 


.84438 


.84931 


.85414 


48 


.81838 


.82380 


.82911 


.83432 


.83944 


.84446 


.84939 


.85422 


52 


.81847 


.82388 


.82920 


.83441 


.83952 


.84454 


.84947 


.85430 


56 


.81857 


.82397 


.82928 


.83449 


.83961 


.84463 


.84955 


.85438 


s 


7A Ijm 7 h 18 m 


7 h 22 7 h 26 


7 h 3O m ? h 3 jm 


7 h Sgm 7 h 2 





9.81866 


9.82406 


9.82937 


9.83458 


9.83969 


9.84471 


9.84963 


9.85446 


4 


.81875 


.82415 


.82946 


.83467 


.83978 


.84479 


.84971 


.85454 


8 


.81884 


.82424 


.82955 


.83475 


.83986 


.84488 


.84979 


.85462 


12 


.81893 


.82433 


.82963 


.83484 


.83995 


.84496 


.84988 


.85470 


'16 


.81902 


.82442 


.82972 


.83492 


.84003 


.84504 


.84996 


.85478 


20 


9.81911 


9.82451 


9.82981 


9.83501 


9.84011 


9.84512 


9.85004 


9.85486 


24 


.81920 


.82460 


.82990 


.83510 


.84020 


.84521 


.85012 


.85494 


28 


.81929 


.82469 


.82998 


.83518 


.84028 


.84529 


.85020 


.85502 


32 


.81938 


.82478 


.83007 


.83527 


.84037 


.84537 


.85028 


.85510 


36 


.81947 


.82487 


.83016 


.83535 


.84045 


.84545 


.85036 


.85518 


40 


9.81956 


9.82495 


9.83025 


9.83544 


9.84054 


9.84554 


9.85044 


9.85526 


44 


.81965 


.82504 


.83033 


.83552 


.84062 


.84562 


.85052 


.85534 


48 


.81975 


.82513 


.83042 


.83561 


.84070 


.84570 


.85061 


.85542 


52 


.81984 


.82522 


.83051 


.83570 


.84079 


.84578 


.85069 


.85550 


56 


.81993 


.82531 


.83059 


.83578 


.84087 


.84587 


.85077 


.85557 


s 


7* 15 m 7* 19 


7 h 23 m 7 h 27 


7 h 31 m 7 h 35 m 


7 h ggm 7 h jlfim 





9.82002 


9.82540 


9.83068 


9.83587 


9.84096 


9.84595 


9.85085 


9.85565 


4 


.82011 


.82549 


.83077 


.83595 


.84104 


.84603 


.85093 


.85573 


8 


.82020 


.82558 


.83086 


.83604 


.84112 


.84611 


.85101 


.85581 


12 


.82029 


.82567 


.83094 


.83612 


.84121 


.84620 


.85109 


.85589 


16 


.82038 


.82575 


.83103 


.83621 


.84129 


.84628 


.85117 


.85597 


20 


9.82047 


9.82584 


9.83112 


9.83630 


9.84138 


9.84636 


9.85125 


9.85605 


24 


.82056 


.82593 


.83120 


.83638 


.84146 


.84644 


.85133 


.85613 


28 


.82065 


.82602 


.83129 


.83647 


.84154 


.84653 


.85141 


.85621 


32 


.82074 


.82611 


.83138 


.83655 


.84163 


.84661 


.85149 


.85629 


36 


.82083 


.82620 


.83147 


.83664 


.84171 


.84669 


.85158 


.85637 


40 


9.82092 


9.82629 


9.83155 


9.83672 


9.84179 


9.84677 


9.85166 


9.85645 


44 


.82101 


.82638 


.83164 


.83681 


.84188 


.84685 


.85174 


.85653 


48 


.82110 


.82646 


.83173 


.83689 


.84196 


.84694 


.85182 


.85660 


52 


.82119 


.82655 


.83181 


.83698 


.84205 


.84702 


.85190 


.85668 


56 


.82128 


.82664 


.83190 


.83706 


.84213 


.84710 


.85198 


.85676 



276 



Table 10. Hayersine Table 



s 


7* 44 ?h ^8 


7 h 52 7 h 56 m 


8 h 8 h 4 m 


8 h 8 8* 12 




Hav. 


Hav. 


Hav. 


Hav. 


Hav. 


Hav. 


Hav. 


Hav. 





'.(..Sf.liM 


9.86153 


9.86613 


9.87064 


9.87506 


9.87939 


9.88364 


9.88780 


4 


.85692 


.86161 


.86621 


.87072 


.87513 


.87947 


.88371 


.88787 


8 


.85700 


.86169 


.86628 


.87079 


.87521 


.87954 


.88378 


.88793 


12 


.85708 


.86176 


.86636 


.87086 


.87528 


.87961 


.88385 


.88800 


16 


.85716 


.86184 


.86643 


.87094 


.87535 


.87968 


.88392 


.88807 


20 


9.85724 


9.86192 


9.86651 


9.87101 


9.87543 


9.87975 


9.88399 


9.88814 


24 


.85731 


.86200 


.86659 


.87109 


.87550 


.87982 


.88406 


.88821 


28 


.85739 


.86207 


.86666 


.87116 


.87557 


.87989 


.88413 


.88828 


32 


.85747 


.86215 


.86674 


.87124 


.87564 


.87996 


.88420 


.88835 


36 


.85755 


.86223 


.86681 


.87131 


.87572 


.88004 


.88427 


.88841 


40 


9.85763 


9.86230 


9.86689 


9.87138 


9.87579 


9.88011 


9.88434 


9.88848 


44 


.85771 


.86238 


.86696 


.87146 


.87586 


.88018 


.88441 


.88855 


48 


.85779 


.86246 


.86704 


.87153 


.87593 


.88025 


.88448 


.88862 


52 


.85787 


.86254 


.86712 


.87161 


.87601 


.88032 


.88455 


.88869 


56 


.85794 


.86261 


.86719 


.87168 


.87608 


.88039 


.88462 


.88876 


s * 


7 h 46 m 7 h A9 m 


7 h 53 7 h 57 m 


gh im $h gm 


8 h 9 m 8 h IS" 1 





9.85802 


9.86269 


9.86727 


9.87175 


9.87615 


9.88046 


9.88469 


9.88882 


4 


.85810 


.86277 


.86734 


.87183 


.87623 


.88053 


.88476 


.88889 


8 


.85818 


.86284 


.86742 


.87190 


.87630 


.88061 


.88483 


.88896 


12 


.85826 


.86292 


.86749 


.87198 


.87637 


.88068 


.88490 


.88903 


16 


.85834 


.86300 


.86757 


.87205 


.87644 


.88075 


.88496 


.88910 


20 


9.85841 


9.86307 


9.86764 


9.87212 


9.87652 


9.88082 


9.88503 


9.88916 


24 


.85849 


.86315 


.86772 


.87220 


.87659 


.88089 


.88510 


.88923 


28 


.85857 


.86323 


.86780 


.87227 


.87666 


.88096 


.88517 


.88930 


32 


.85865 


.86331 


.86787 


.87235 


.87673 


.88103 


.88524 


.88937 


36 


.85873 


.86338 


.86795 


.87242 


.87680 


.88110 


.88531 


.88944 


40 


9.85881 


9.86346 


9.86802 


9.87249 


9.87688 


9.88117 


9.88528 


9.88950 


44 


.85888 


.86354 


.86810 


.87257 


.87695 


.88124 


.88545 


.88957 


48 


.85896 


.86361 


.86817 


.87264 


.87702 


.88131 


.88552 


.88964 


62 


.85904 


.86369 


.86825 


.87271 


.87709 


.88139 


.88559 


.88971 


56 


.85912 


.86377 


.86832 


.87279 


.87717 


.88146 


.88566 


.88978 


8 


7/> 4#m 7 h 50 m 


7 h 54 m 7 h 58" 1 


8 h 2 8 h 6 


8 h io m 8^ 14 m 





9.85920 


9.86384 


9.86840 


9.87286 


9.87724 


9.88153 


9.88573 


9.88984 


4 


.85928 


.86392 


.86847 


.87294 


.87731 


.88160 


.88580 


.88991 


8 


.85935 


.86400 


.86855 


.87301 


.87738 


.88167 


.88587 


.88998 


12 


.85943 


.86407 


.86862 


.87308 


.87745 


.88174 


.88594 


.89005 


16 


.85951 


.86415 


.86870 


.87316 


.87753 


.88181 


.88600 


.89012 


20 


9.85959 


9.86423 


9.86877 


9.87323 


9.87760 


9.88188 


9.88607 


9.89018 


24 


.85967 


.86430 


.86885 


.87330 


.87767 


.88195 


.88614 


.89025 


28 


.85974 


.86438 


.86892 


.87338 


.87774 


.88202 


.88621 


.89032 


32 


.85982 


.86446 


.86900 


.87345 


.87782 


.88209 


.88628 


.89039 


36 


.85990 


.86453 


.86907 


.87352 


.87789 


.88216 


.88635 


.89045 


40 


9.85998 


9.86461 


9.86915 


9.87360 


9.87796 


9.88223 


9.88642 


9.89052 


44 


.86006 


.86468 


.86922 


.87367 


.87803 


.88230 


.88649 


.89059 


48 


.86013 


.86476 


.86930 


.87374 


.87810 


.88237 


.88656 


.89066 


52 


.86021 


.86484 


.86937 


.87382 


.87818 


.88244 


.88663 


.89072 


56 


.86029 


.86491 


.86945 


.87389 


.87825 


.88252 


.88670 


.89079 


s 


7* 47 m 7* 51 m 


7* 55 m 7 h 59 


8 h 3 m 8* 7 m 


8 h Hm 8 h 15 m 





9.86037 


9.86499 


9.86952 


9.87396 


9.87832 


9.88259 


9.88677 


9.89086 


4 


.86045 


.86507 


.86960 


.87404 


.87839 


.88266 


.88683 


.89093 


8 


.86052 


.86514 


.86967 


.87411 


.87846 


.88273 


.88690 


.89099 


12 


.86060 


.86522 


.86975 


.87418 


.87853 


.88280 


.88697 


.89106 


16 


.86068 


.86529 


.86982 


.87426 


.87861 


.88287 


.88704 


.89113 


20 


9.86076 


9.86537 


9.86990 


9.87433 


9.87868 


9.88294 


9.88711 


9.89120 


24 


.86083 


.86545 


.86997 


.87440 


.87875 


.88301 


.88718 


.89126 


28 


.86091 


.86552 


.87004 


.87448 


.87882 


.88308 


.88725 


.89133 


32 


.86099 


.86560 


.87012 


.87455 


.87889 


.88315 


.88732 


.89140 


36 


.86107 


.86568 


.87019 


.87462 


.87896 


.88322 


.88739 


.89147 


40 


9.86114 


9.86575 


9.87027 


9.87470 


9.87904 


9.88329 


9.88745 


9.89153 


44 


.86122 


.86583 


.87034 


.87477 


.87911 


.88336 


.88752 


.89160 


48 


.86130 


.86590 


.87042 


.87484 


.87918 


.88343 


.88759 


.89167 


62 


.86138 


.86598 


.87049 


.87492 


.87925 


.88350 


.88766 


.89174 


56 


.86145 


.86606 


.87057 


.87499 


.87932 


.88357 


.88773 


.89180 


60 


9.86153 


9.86613 


9.87064 


9.87506 


9.87939 


9.88364 


9.88780 


9.89187 



Table 10. Haversine Table 



277 



s 


8 h 16 8 h 20 m 


gh 24 m 8* 28 


8 h 32 m gh 36 m 


8 h 40 8 h 44 m 




Hav. 


Hav. 


Hav. 


Hav. 


Hav. 


Hav. 


Hav. 


Hav. 





9.89187 


9.89586 


(.89976 


9.90358 


9.90732 


9.91098 


9.91455 


9.91805 


4 


.89194 


.89592 


.89983 


.90365 


.90738 


.91104 


.91461 


.91810 


8 


.89200 


.89599 


.89989 


.90371 


.90744 


.91110 


.91467 


.91816 


12 


.89207 


.89606 


.89995 


.90377 


.90751 


.91116 


.91473 


.91822 


16 


.89214 


.89612 


.90002 


.90383 


.90757 


.91122 


.91479 


.91828 


20 


9.89221 


9.89619 


9.90008 


9.90390 


9.90763 


9.91128 


9.91485 


9.91833 


24 


.89227 


.89625 


.90015 


.90396 


.90769 


.91134 


.91490 


.91839 


28 


.89234 


.89632 


.90021 


.90402 


.90775 


.91140 


.91496 


.91845 


32 


.89241 


.89638 


.90028 


.90409 


.90781 


.91146 


.91502 


.91851 


36 


.89247 


.89645 


.90034 


.90415 


.90787 


.91152 


.91508 


.91856 


40 


9.89254 


9.89651 


9.90040 


9.90421 


9.90794 


9.91158 


9.91514 


9.91862 


44 


.89261 


.89658 


.90047 


.90428 


.90800 


.91164 


.91520 


.91868 


48 


.89267 


.89665 


.90053 


.90434 


.90806 


.91170 


.91526 


.91874 


52 


.89274 


.89671 


.90060 


.90440 


.90812 


.91176 


.91532 


.91879 


56 


.89281 


.89678 


.90066 


.90446 


.90818 


.91182 


.91537 


.91885 


s 


gh 17 8* 21 m 


8h 25 m 8* 29 


gh 33 m g h S7 m 


gh 4Jm gh 45 





9.89287 


9.89684 


9.90072 


9.90452 


9.90824 


9.91188 


9.91543 


9.91891 


4 


.89294 


.89691 


.90079 


.90459 


.90830 


.91194 


.91549 


.91896 


8 


.89301 


.89697 


.90085 


.90465 


.90836 


.91200 


.91555 


.91902 


12 


.89308 


.89704 


.90092 


.90471 


.90843 


.91206 


.91561 


.91908 


16 


.89314 


.89710 


.90098 


.90478 


.90849 


.91212 


.91567 


.91914 


20 


9.89321 


9.89717 


9.90104 


9.90484 


9.90855 


9.91218 


9.91573 


9.91919 


24 


.89328 


.89723 


.90111 


.90490 


.90861 


.91224 


.91578 


.91925 


28 


.89334 


.89730 


.90117 


.90496 


.90867 


.91230 


.91584 


.91931 


32 


.89341 


.89736 


.90124 


.90503 


.90873 


.91236 


.91590 


.91936 


36 


.89348 


.89743 


.90130 


.90509 


.90879 


.91242 


.91596 


.91942 


40 


9.89354 


9.89749 


9.90136 


9.90515 


9.90885 


9.91248 


9.91602 


9.91948 


44 


.89361 


.89756 


.90143 


.90521 


.90892 


.91254 


.91608 


.91954 


48 


.89368 


.89763 


.90149 


.90527 


.90898 


.91260 


.91613 


.91959 


52 


.89374 


.89769 


.90156 


.90534 


.90904 


.91265 


.91619 


.91965 


56 


.89381 


.89776 


.90162 


.90540 


.90910 


.91271 


.91625 


.91971 


s 


8 h 18 m 8^ 22 


8 h 26 8>> 30 


8* 34 m 8 h 38 


gh Jgm gh ^ffn 





9.89387 


9.89782 


9.90168 


9.90546 


9.90916 


9.91277 


9.91631 


9.91976 


4 


.89394 


.89789 


.90175 


.90552 


.90922 


.91283 


.91637 


.91982 


8 


.89400 


.89795 


.90181 


.90559 


.90928 


.91289 


.91643 


.91988 


12 


.89407 


.89802 


.90187 


.90565 


.90934 


.91295 


.91648 


.91993 


16 


.89414 


.89808 


.90194 


.90571 


.90940 


.91301 


.91654 


.91999 


20 


9.89421 


9.89815 


9.90200 


9.90577 


9.90946 


9.91307 


9.91660 


9.92005 


24 


.89427 


.89821 


.90206 


.90584 


.90952 


.91313 


.91666 


.92010 


28 


.89434 


.89828 


.90213 


.90590 


.90958 


.91319 


.91672 


.92016 


32 


.89441 


.89834 


.90219 


.90596 


.90965 


.91325 


.91677 


.92022 


36 


.89447 


.89840 


.90225 


.90602 


.90971 


.91331 


.91683 


.92027 


40 


9.89454 


9.89847 


9.90232 


9.90608 


9.90977 


9.91337 


9.91689 


9.92033 


44 


.89460 


.89853 


.90238 


.90615 


.90983 


.91343 


.91695 


.92039 


48 


.89467 


.89860 


.90244 


.90621 


.90989 


.91349 


.91701 


.92044 


62 


.89474 


.89866 


.90251 


.90627 


.90995 


.91355 


.91706 


.92050 


66 


.89480 


.89873 


.90257 


.90633 


.91001 


.91361 


.91712 


.92056 


s 


8 h 19 m 8 h 23 


gh 27 m 8* 31 m 


8* 35 m 8* 39 m 


8 h 43 8 h 4? m 





9.89487 


9.89879 


9.90264 


9.90639 


9.91007 


9.91367 


9.91718 


9.92061 


4 


.89493 


.89886 


.90270 


.90646 


.91013 


.91372 


.91724 


.92067 


8 


.89500 


.89892 


.90276 


.90652 


.91019 


.91378 


.91730 


.92073 


12 


.89507 


.89899 


.90282 


.90658 


.91025 


.91384 


.91735 


.92078 


16 


.89513 


.89905 


.90289 


.90664 


.91031 


.91390 


.91741 


.92084 


20 


9.89520 


9.89912 


9.90295 


9.90670 


9.91037 


9.91396 


9.91747 


9.92090 


24 


.89527 


.89918 


.90301 


.90676 


.91043 


.91402 


.91753 


.92095 


28 


.89533 


.89925 


.90308 


.90683 


.91049 


.91408 


.91758 


.92101 


' 32 


.89540 


.89931 


.90314 


.90689 


.91055 


.91414 


.91764 


.92107 


36 


.89546 


.89938 


.90320 


.90695 


.91061 


.91420 


.91770 


.92112 


40 


9.89553 


9.89944 


9.90327 


9.90701 


9.01067 


9.91426 


9.91776 


9.92118 


44 


.89559 


.89950 


.90333 


.90707 


.91074 


.91432 


.91782 


.92124 


48 


.89566 


.89957 


.90339 


.90714 


.91080 


.91437 


.91787 


.92129 


52 


.89573 


.89963 


.90346 


.90720 


.91086 


.91443 


.91793 


.92135 


56 


.89579 


.89970 


.90352 


.90726 


.91092 


.91449 


.91799 


.92140 


60 


9.89586 


9.89976 


9.90358 


9.90732 


9.19098 


9.91455 


9.91805 


9.92146 



278 



Table 10. Haversine Table 



s 


gh j^gm 8>> 52 


8 h 56 m 9* O m 


9* 4 m 9*8 


gh Igm gh Jgm 




Hav. 


Hav. 


Hav. 


Hav. 


Hav. 


Hav. 


Hav. 


Hav. 





9.92146 


9.92480 


9.92805 


3.93123 


9.93433 


9.93736 


9.94030 


9.94318 


4 


.92152 


.92485 


.92811 


.93128 


.93438 


.93741 


.94035 


.94322 


8 


.92157 


.92491 


.92816 


.93134 


.93443 


.93746 


.94040 


.94327 


12 


.92163 


.92496 


.92821 


.93139 


.93448 


.93751 


.94045 


.94332 


16 


.92169 


.92502 


.92827 


.93144 


.93454 


.93755 


.94050 


.94336 


20 


9.92174 


9.92507 


9.92832 


9.93149 


9.93459 


9.93760 


9.94055 


9.94341 


24 


.92180 


.92512 


.92837 


.93154 


.93464 


.93765 


.94059 


.94346 


28 


.92185 


.92518 


.92843 


.93160 


.93469 


.93770 


.94064 


.94351 


32 


.92191 


.92523 


.92848 


.93165 


.93474 


.93775' 


.94069 


.94355 


86 


.92197 


.92529 


.92853 


.93170 


.93479 


.93780 


.94074 


.94360 


40 


9.92202 


9.92534 


9.92859 


9.93175 


9.93484 


9.93785 


9.94079 


9.94365 


44 


.92208 


.92540 


.92864 


.93181 


.93489 


.93790 


.94084 


.94369 


48 


.92213 


.92545 


.92869 


.93186 


.93494 


.93795 


.94088 


.94374 


62 


.92219 


.92551 


.92875 


.93191 


.93499 


.93800 


.94093 


.94379 


56 


.92225 


.92556 


.92880 


.93196 


.93504 


.93805 


.94098 


.94383 


s 


8 h 49 8 h 63 


gh 5?m Qh im 


Qh gm. Q h gm 


gh ism Qh ijm 





9.92230 


9.92562 


9.92885 


9.93201 


9.93509 


9.93810 


9.94103 


9.94388 


4 


.92236 


.92567 


.92891 


.93207 


.93515 


.93815 


.94108 


.94393 


8 


.92241 


.92573 


.92896 


.93212 


.93520 


.93820 


.94112 


.94398 


12 


.92247 


.92578 


.92901 


.93217 


.93525 


.93825 


.94117 


.94402 


16 


.92253 


.92584 


.92907 


.93222 


.93530 


.93830 


.94122 


.94407 


20 


9.92258 


9.92589 


9.92912 


9.93227 


9.93535 


9.93835 


9.94127 


9.94412 


^4 


.92264 


.92594 


.92917 


.93232 


.93540 


.93840 


.94132 


.94416 


28 


.92269 


.92600 


.92923 


.93238 


.93545 


.93845 


.94137 


.94421 


32 


.92275 


.92605 


.92928 


.93243 


.93550 


.93849 


.94141 


.94426 


S6 


.92280 


.92611 


.92933 


.93248 


.93555 


.93854 


.94146 


.94430 


40 


9.92286 


9.92616 


9.92939 


9.93253 


9.93560 


9.93859 


9.94151 


9.94435 


44 


.92292 


.92622 


.92944 


.93258 


.93565 


.93864 


.94156 


.94440 


48 


.92297 


.92627 


.92949 


.93264 


.93570 


.93869 


.94161 


.94444 


52 


.92303 


.92633 


.92955 


.93269 


.93575 


.93874 


.94165 


.94449 


66 


.92308 


.92638 


.92960 


.93274 


.93580 


.93879 


.94170 


.94454 


s 


gh 5^ gh S4 m 


8 h 58 & 2 


9 h 6 & 10 


Qh IJfn Qh 18 m 





9.92314 


9.92043 


9.92965 


9.93279 


9.93585 


9.93884 


9.94175 


9.94458 


4 


.92319 


.92649 


.92970 


.93284 


.93590 


.93889 


.94180 


.94463 


8 


.92325 


.92654 


.92975 


.93289 


.93595 


.93894 


.94184 


.94468 


12 


.92330 


.92660 


.92981 


.93295 


.93600 


.93899 


.94189 


.94472 


16 


.92336 


.92665 


.92986 


.93300 


.93605 


.93904 


.94194 


.94477 


20 


9.92342 


9.92670 


9.92992 


9.93305 


9.93611 


9.93908 


9.94199 


9.94482 


24 


.92347 


.92676 


.92997 


.93310 


.93616 


.93913 


.94204 


.94486 


28 


.92353 


.92681 


.93002 


.93315 


.93621 


.93918 


.94208 


.94491 


32 


.92358 


.92687 


.93007 


.93320 


.93626 


.93923 


.94213 


.94496 


36 


.92364 


.92692 


.93013 


.93326 


.93631 


.93928 


.94218 


.94500 


40 


9.92369 


9.92698 


9.93018 


9.93331 


9.93636 


9.93933 


9.94223 


9.94505 


44 


.92375 


.92703 


.93023 


.93336 


.93641 


.93938 


.94227 


.94509 


48 


.92380 


.92708 


.93029 


.93341 


.93646 


.93943 


.94232 


.94514 


62 


.92386 


.92714 


.93034 


.93346 


.93651 


.93948 


,94237 


.94519 


66 


.92391 


.92719 


.93039 


.93351 


.93656 


.93952 


.94242 


.94523 


s 


gh Sim 8 h 55 m 


8 h 59 & S" 1 


Qh j. gh Jjm 


gh ism Qh IQm 





9.92397 


9.92725 


9.93044 


9.93356 


9.93661 


9.93957 


9.94246 


9.94528 


4 


.92402 


.92730 


.93050 


.93362 


.93666 


.93962 


.94251 


.94533 


8 


.92408 


.92735 


.93055 


.93367 


.93671 


.93967 


.94256 


.94537 


12 


.92413 


.92741 


.93060 


.93372 


.93676 


.93972 


.94261 


.94542 


16 


.92419 


.92746 


.93065 


.93377 


.93681 


.93977 


.94265 


.94546 


20 


9.92425 


9.92751 


9.93071 


9.93382 


9.93686 


9.93982 


9.94270 


9.94551 


24 


.92430 


.92757 


.93076 


.93387 


.93691 


.93987 


.94275 


.94556 


28 


.92436 


.92762 


.93081 


.93392 


.93696 


.93991 


.94280 


.94560. 


S2 


.92441 


.92768 


.93086 


.93397 


.93701 


.93996 


.94284 


.94565 


S6 


.92447 


.92773 


.93092 


.93403 


.93706 


.94001 


.94289 


.94570 


40 


9.92452 


9.92778 


9.93097 


9.93408 


9.93711 


9.94006 


9.94294 


9.94574 


44 


.92458 


.92784 


.93102 


.93413 


.93716 


.94011 


.94299 


.94579 


48 


.92463 


.92789 


.93107 


.93418 


.93721 


.94016 


.94303 


.94583 


62 


.92469 


.92794 


.93113 


.93423 


.93726 


.94021 


.94308 


.94